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Within the category of graded manifolds, the most important seem to be the non-negatively graded manifolds. If the Grassmann parity of the coordinates coincides with the weight (mod 2), then we have a {{formula:1221ece6-04ff-4f09-b4d3-c6c2265c77d9}} -manifold (cf. {{cite:df09bb101395b91e8030276568b66bd56013b387}}, {{cite:39a4e5331a29e371e20accd72773e434b04f6c1d}}). In the purely even setting, Grabowski and Rotkiewicz {{cite:d63247ada7c6b626acfe82c709ddcc56037433d4}} define what they referred to as graded bundles, being a particular class of non-negatively graded manifolds. They showed that a graded bundle, understood as a manifold with a non-negative grading on its structure sheaf, is equivalent to a manifold equipped with a smooth action of the multiplicative monoid {{formula:c2743259-dcb9-4f19-a384-fcbc9e3791b2}} of reals. Such actions they call homogeneity structures (we will be more precise shortly). As this action reduced to {{formula:508f6992-2bb3-42ed-a8aa-30738be09367}} is just generated by the weight vector field, it means, actually by definition, that the weight vector field is {{formula:b05143f7-59d2-4873-abc5-90aa2e0ea8a8}} -complete.
| r | 59d6409a502d5c02ee4ba900dc0f357f |
RetinaNet {{cite:7f5bfdcbaf885008b244a501c1007aea8f4b134d}} takes into account the data driven property that allows the network to focus on “hard” samples for improved accuracy. The easy to adapt backbones for feature extraction at the beginning of the network provides the opportunity to experiment with deeper and varied architectures such as ResNet50, and ResNet101 for RetinaNet and 53 layered Darknet53 backbone for YOLOv3 and YOLOv4 architecture. To tackle the different aspect ratio problem, for both one stage networks, optimal anchor boxes {{cite:2339703cd8693b806de70693b3c7299f244b6657}} are searched and pre-defined for the provided data to tackle large variance of scale and aspect ration of boxes. Table REF shows the hyperparameter used by each of the object detection methods for the detection task.
| m | 0f749c547ed9559962eb9caec473bf72 |
In this paper, three problems are discussed: the damping of primordial electric fields, electric solitons and their effects on acoustic oscillations.
As calculated in relevant cosmological monographs {{cite:19dbeaf5c63a4e1b510db7036f1db44f3e41f4a7}}, the effects on CMB from electromagnetic fields are often ignored,
although, in plasma, such effects always play extremely significant roles.
| d | 61df0a244ddcf69ea6ef2bd45a635040 |
NGC 5322 is also a core-depleted galaxy with a large deficit of stellar mass in the core {{cite:a664d52309fa10495d8f1c9cd7b113457daf985e}}. Such cores can result from dry mergers of galaxies. NGC 5322 lacks rich ISM as evidenced from non-detection of both molecular hydrogen {{cite:356fb1fbbb921561767e929d114b3e173cb3fbfc}} and neutral hydrogen emission {{cite:2c982a0497255965fd5470b37bb91af8fb31bc99}}, {{cite:892d1e6c76b25adb5fcdcdc19ab4e2cdbb61e06a}}. Only narrow neutral hydrogen absorption has been detected {{cite:892d1e6c76b25adb5fcdcdc19ab4e2cdbb61e06a}}. This absorption is likely to be originating in the edge-on dusty kpc-scale disk detected in the HST image.
| d | d8ab9f95bd72a75da43dbf2cf9f5f261 |
However, there are many limitations of this work.
First, the nonparametric framework analyzes properties of the empirical risk minimizer, e.g., convergence rate, sample complexity, etc., but doesn't provide guidance to practical optimization and how to find them.
Second, the proposed localized separation condition, like Tsybakov's noise condition, is of theoretical interest and cannot be verified for real data such as natural images. Similarly, this work only considers the 0-1 loss, which is also out of theoretical interest rather than practical relevance since it's the most natural and fundamental classification loss.
For future work, it is intriguing to study how DNN classifiers perform under popular surrogate losses such as hinge loss or cross-entropy.
Our numerical experiments conducted using cross-entropy indicate similar results may also hold for empirical surrogate loss minimizers.
Our theory can be further strengthened if we could relax the structural requirement for the DNN family to be more general, e.g, fully-connected, or extend the analysis to popular networks such as convolutional neural networks {{cite:71eb08d0fd060afd94433a6c45bec1d1acec8530}}, residual neural networks {{cite:b21b999d0b5f92ba8f3ef0881e56187b17bd31fb}}, and Transformers {{cite:67055b5b8e93f5e06ef10fd846dde564f0a55011}}.
| d | b3dbccb4305500d14f194bff2c4ebebc |
The idea of using a subset of the matched asymptotic equations (constraint equations) to learn about the backreacted description of a brane configuration in some background lies at the “soul” of the blackfold approach. In the blackfold approach, the constraint equations, which are dubbed blackfold equations, provide a {{formula:4ce41a9b-ec8b-4bcf-859f-13e9fac33259}} -dimensional effective worldvolume description of the dynamics of a {{formula:04808d2d-da40-40a6-8a6c-8b03d658ee6e}} -brane solution. When viewed in isolation, the blackfold equations are only the necessary conditions for a matched asymptotic solution. As such, statements made using the blackfold approach are only truly conclusive when a regular perturbative solution in MAE can be constructed. The claim that the blackfold equations guarantee a regular matched asymptotic solution is dubbed the blackfold conjecture and this conjecture is baked into many of the blackfold applications including that of {{cite:741cb347b6ab056d06b280709b72df8909766236}}. The conjecture is correct in the fluid-gravity correspondence {{cite:cf60422cbf0395a4340410730ea755ef4557bcda}} in AdS/CFT and has been proven at leading order in the MAE expasion in pure Einstein gravity in flat space {{cite:baf8f2ae5bb175584b2f152e997bf1875d9e4b0b}}. It is not known, however, if it is generally valid in supergravity with generic asymptotics. In particular, it has not been proved to be valid in the context of the highly-nontrivial configurations of anti-branes in the warped, fluxed KS throat. It is the lack of this proof that motivates the study in this paper.
| i | aa8edaf234cbf0068104899e8c405149 |
For Eq. (5), we utilize the D3Q15 lattice structure {{cite:f2ecaafacbec3d89861f276967cd4cdbbc1829ed}}, where the weight coefficient is given by {{formula:41ae9053-1b1e-492d-bd78-5f06bf804531}} , {{formula:041a07ba-86bf-44ca-918f-722a38666b57}} , {{formula:70984eb6-ea6d-446b-813a-fdb3978d6cce}} , {{formula:c7f6e5b0-41c6-474b-8ef5-4adfe6be6c14}} , the discrete velocity {{formula:4aff3052-add0-49e3-a73a-bd2cfc25e77e}} is defined as
{{formula:cf66d8a6-f960-4ffd-9dad-b8e3cfd107c2}}
| m | 004a7f776ea2a4ecb497e7be8d13998f |
From the perspective of the user-item interaction graph, the individual interaction history is equivalent to the first-order connectivity of the user. Thus, a natural extension is to mine the higher-order connectivity from the user-item graph structure. For example, the second-order connectivity of a user consists of similar users who have co-interacted with the same items. Fortunately, with the development and success of graph convolutional networks (GCNs) for modeling graph structure data in various machine learning areas, it recently became popular to adopt GCNs for CF {{cite:100c8315dcc179b561a68d3f595d7e53be8b8ea6}}, {{cite:8c34a1a7cf880d19d9f403b19481ef8377d5b934}}, {{cite:3ce484c0dcdda8f1b2c05437ce877eed6b35ba24}}, {{cite:36bcb105598f7814e5ebce4be393446743b69b1a}}, {{cite:60eeae2d1f0f6b3156255930420c670d9397e355}}, {{cite:100c8315dcc179b561a68d3f595d7e53be8b8ea6}}, {{cite:6bd9be82a318add3df6ebb7cad92a4f7a57f7514}}, {{cite:44be0af29ee4e3ee3bdf167f4c97ac4fcc17de87}}.
| m | 50c16578259b77527689f17729f46f6f |
By going back to the original analogy with genes put forward by Richard Dawkins {{cite:4d7508c555371cf8913661b381410e7665bb3fce}}, we investigated the relation between the occurrence frequency of scientific memes and the degree to which they propagate along the citation graph. We found that scientific memes are indeed governed by a surprisingly simple relationship of these two factors.
This is formalized by the meme score — a metric to characterize and identify scientific memes — defined as the product of the frequency of occurrence and the propagation score.
| d | 04107b700ea39e154c81b49697581879 |
Influence of backbone networks. Besides the simple Conv-64F, we also use other deeper feature extractors to evaluate our model, i.e., ResNet12 and WRN-28-10. We compared other state-of-the-art methods that using these deeper feature extractors, including Dynamic-Net {{cite:fb34d8978bb4c333d8f6870f74c497ffbe6e61d6}}, SNAIL {{cite:6c8e6baf1d1f3307a3460310f6faf2b06d0b8269}}, TPN {{cite:7adc894c160401459ecee519e92241977bdd6652}}, MAML+L2F {{cite:253078e53dbd049c420e1517a66aa273fc2b2e1d}}, Qiao {{cite:dba52c1f33bdfcf0bd49fce85e6d528c0252e1fc}}, LEO {{cite:3e00532f8d71d31a888acdd9078f7c54861c21b2}}, Fine-tuning {{cite:a99002f7af8a202dbfcbfcc57e87e89b8cda8f40}} and LEO+L2F {{cite:253078e53dbd049c420e1517a66aa273fc2b2e1d}}. When using deeper feature extractors, the accacy of MATANet reaches 60.13% and 62.43% for the 5-way 1-shot task, 75.42% and 79.13% for the 5-shot task, by using ResNet12 and WRN-28-10, repectively. Moreover, when using same deeper feature exactor, our MATANet outperforms all other methods under both 5-way 1-shot and 5-way 5-shot few-shot learning settings (see Table 4).
| d | 16fce1975ee9ebd0e346f968f32f14bf |
Despite its similarity to standard ODQA tasksHistorically, open-domain QA meant “QA on any domain/topic”. More recently, the term has been restricted to “retrieval on a large pile of corpus” {{cite:99a741aaa3c42f9e51b42c2cd53bb99b1e8a0783}}, so “open-retrieval QA” seems a better term here. However, to follow the recent terminology in the QA community, we still use “open-domain QA” throughout this paper., i.e., both requiring finding evidence paragraphs for inferring answers, the Book QA has certain unique challenges {{cite:cdb74370e9d576b35e4afedcadef3ed01204c93a}}:
(1) the narrative writing style of book stories differs from the formal texts in Wikipedia and news, which demands a deeper understanding capability. The flexible writing styles from different genres and authors make the challenge severe;
(2) the passages that depict related book plots and characters share more semantic similarities than the Wikipedia articles, which increases confusion in finding the correct evidence to answer a question;
(3) the free-form nature of the answers necessitates the summarization ability from the narrative plots; (4) the free-form answers make it hard to obtain fine-grained supervision at passage or span levels;
and finally (5) different paragraphs usually have logical relations among them.We consider Challenge (5) more like an opportunity than challenges, and leave its investigation to future work.
| i | 96cd3111088f22c9b926d8be0053cb4d |
An important feature of convolutional networks is their equivariance (consistency) with respect to the symmetry transformations of the input data {{cite:83ee2cd2fd2b19e9f867d4cc1d1a6700aae05053}}, {{cite:40ac02a5268a1c49292e585a183f971aaea4eb4c}}. Equivariance guarantees that exactly the same filters are applied to each part of the input image regardless of position and that the network can detect any given object equally well regardless of its location respecting data symmetry properties.
| i | 00dc99588ff0887dfc901891fea88ddb |
Table REF lists the comparison between the proposed HSTGCNN model and the latest state-of-the-art methods using ROC AUC. The ten methods are Frame-Pred {{cite:72afaa1150ce8dd772c1872be33e0ae14aba41a0}}, MPED-RNN {{cite:2bb8e7bdf7f83a9a0b28b1d5a742473c8aae6649}}, w/Mem {{cite:f6c8239c998ba6855b8a1d1be0d4a2fabb23a572}}, ST-GCAE {{cite:50190288c9017c2a540ff1fee0fd05d37c671e50}}, Multi-timescale {{cite:26444be424e07c623f8923165d110ac44490884f}}, PoseCVAE {{cite:5731b9599e0077ad9fca7ca3447d24baf1e1465c}}, LSA {{cite:1f4c83f3b7618319d3f1cefdc5713130df68b586}}, Ano-Graph {{cite:2d86090a48a598ed5b0e7a932030ff0fc2f07de5}}, AnomalyNet {{cite:fa37a6657cc02f5736ecee101f492c680c10d232}}, and Normal Graph {{cite:4208e17e26cf6a84390d47fe78016295375cbf2c}}, some of them integrate a model focusing on appearance and motion with others dealing with the trajectories of human skeletons. From experiments, we can conclude that HSTGCNN outperforms the ten methods mentioned above on four public datasets, including Human-Related (HR) and original datasets. Although there are anomalies unrelated to humans in the video segment of the dataset, it still achieves the highest frame-level ROC AUC. As a result, compared with methods based on pixel reconstruction {{cite:72afaa1150ce8dd772c1872be33e0ae14aba41a0}},{{cite:fa37a6657cc02f5736ecee101f492c680c10d232}} and {{cite:1f4c83f3b7618319d3f1cefdc5713130df68b586}}, the model shows a stronger robustness. The noises of intensity-based features can be reduced by extracting the features describing human skeletons and motion instead of pixels. By levering a novel structure, our approach achieves advantages over RNNs and improves understanding of the global scenes in contrast to the coarse-grained STGCNN {{cite:4208e17e26cf6a84390d47fe78016295375cbf2c}}, {{cite:2d86090a48a598ed5b0e7a932030ff0fc2f07de5}}. As expected, anomalous events about humans can be correctly detected in different scenes in the ShanghaiTech dataset, as shown in Fig. REF . In addition, skeleton-based sequences and motion-related features are integrated by high-level graph representations to accommodate different resolution datasets. To conclude, effective structures, comprehensive feature representations involving high/low-level embeddings and adjustable weights according to scenes contribute to the significant improvement of the overall accuracy and robustness of the HSTGCNN method.
{{figure:ce7e276c-783f-4272-8ec2-bb285e4ccae5}} | m | 3e69e259f7775e87eb6e4758ff31cf8d |
With the increasing possibility of Urban Air Mobility (UAM) in the recent future, quadrotors' need to navigate through urban high-rises becomes increasingly inevitable. Monocular vision continues to be a widespread perception modality for quadrotors considering payload, portability, and endurance. Hence navigation based on monocular SLAM maps becomes a natural choice. However, these maps tend to be sparse, noisy, and not represented in the scale of the actual world. We can use the popular Visual Inertial SLAM {{cite:84b29f175d923c56476add40b89d2e1c77792a62}} to obtain the quadrotor odometry and consequently bring the maps to metric scale. Yet, uncertainty and sparsity of the point clouds are not entirely alleviated. Thus, it becomes critical to account for this uncertainty during trajectory planning. Unfortunately, existing algorithms for uncertainty-aware planning are not designed to work with systems that rely on monocular vision-based perception. For example, works like {{cite:378655e5991d73c1d207159271ceb5f1c6419638}}, {{cite:9703d102a99ba952d7fdf59c8fd6dd54539ca103}}, either assume that the obstacle geometries are precisely known or rely on depth sensors to obtain obstacle geometries during run-time.
| i | 111084b5d8e671d3a230cd5e4b40382f |
In Table REF , we have summarized the contributions
of stochastic GWs to the fluxes of flavor neutrinos at the Earth.
They are at the level of a few percent. The current neutrino telescopes
are able to detect up to several thousand neutrinos from SN in our
Galaxy {{cite:a7600e6a70cfa9d1f2878acd4495efe3ddfbe443}}. Future detectors, like the Hyper-Kamiokande,
can detect about {{formula:d9a2feee-277d-4c3d-8061-9e3bf425d1f8}} such events {{cite:2f2296c414a55723c76f0f1704486c0d47b834e5}}. Thus,
the interaction with stochastic GWs can results in the change of the
SN neutrinos fluxes by {{formula:f3fa0c25-3ca0-41f4-b24d-17492d01f9b4}} events, in case of the Super-Kamiokande,
and by {{formula:c84f21c9-ff0d-4d7e-83a3-bfb94998db31}} events, for the Hyper-Kamiokande.
| d | 15ec623c37e2ddda2fc7a6b8c5e6bd7e |
Mathematical modeling allows us to analyze various natural phenomena and find optimal solutions. Two pioneers in this area, Kuramoto {{cite:1dd17086a25d86b932739a5380279da65906832c}}, {{cite:d7622f6a9413a6b1eeb15808b23481554925bc48}} and Winfree {{cite:161fbd9cae928e05693eebf94c860eade4915180}}, catalyzed studying mathematical modeling and its applications. Especially, two mathematical models introduced by these two pioneers are pairwise interaction models which can be written in the following form:
{{formula:8fe68844-9f7e-4103-b421-c580dfc0494f}}
| i | 4ff43974b3d1137e409b887b681e8799 |
Deep Convolutional Neural Networks (dcnns) have proved to be useful in highly complex computer vision tasks to identify visually distinguished features of images. The effectiveness of dcnns are also being explored in recent years in document object analysis by various research groups {{cite:71d2112241ff2db861e5d508e9fcf4c363a2c79c}}, {{cite:e7ac3309eee359eec63c814ba0da8968b38c7333}}, {{cite:71922b072f0beaad52e0bc28ab0286c09b919cba}}, {{cite:ffdadcb461c90c1af56dbf1f7173ec038c558416}}. Hao et al. {{cite:71d2112241ff2db861e5d508e9fcf4c363a2c79c}} applied deep learning for detection of tables in pdf documents. They considered heuristic rules to propose regions with table-like structures in the document and then classified them into table or non-table regions using a cnn. However, the shortcomings of using rule to identify table-like structures are not yet overcome completely by this method. Schreiber et al. {{cite:e7ac3309eee359eec63c814ba0da8968b38c7333}} proposed a deep learning based solution for table detection and structure identification, which does not require any assumption about the structure of the tables. They fine-tuned Faster r-cnn {{cite:45784deed016cb2b576b5957136757a7f441992f}} with two different backbone architectures: zfnet {{cite:15fb526e7945f885db80b066321b0fe2adf4c39d}} and vgg-16 {{cite:1805ec1131462761220070f7115db660c6939b2d}} for the table detection task. Gilani et al. {{cite:71922b072f0beaad52e0bc28ab0286c09b919cba}} also used the same approach. However, they applied image transformation by stacking three different distance transformed layers before passing them through Faster r-cnn. This model can deal with only identifying table regions in the documents.
{{figure:5fcda69a-7b89-4503-8dca-71814eb16971}} | m | de3a3938ce7561c9647148a6d1533484 |
where {{formula:5784e335-6b58-4b05-a8e6-33fe800a515b}} is an inter-UAV attractive/repulsive term, {{formula:09b214ec-3e4c-49aa-b6b2-f13064b04c8c}} is a velocity consensus term, {{formula:ac71e68c-5a7a-418c-b6cc-e64ada77bb1f}} is a term defining the individual goal of each UAV. In order to create smooth incentive functions with finite cut-offs, we leverage the bump function {{formula:b265e364-b0fe-41f0-a040-6b11246b2398}} in {{cite:45e05534eacccdff347a823fa46cd6e7d860866c}} and {{formula:59fc9a6e-00e6-435e-a061-0c11a0852072}} -norm in {{cite:f7564eb9cf9dbc8800f614bda23116c4627fa71c}}. The bump function is defined as follows:
{{formula:dc1ab875-1fe9-4213-a1c8-cb9ddf0b5603}}
| m | 1fe1e5a55ad3147745160d5ee71fbd67 |
The first consequence follows from the fact that the scalar field {{formula:914ec7a9-eee2-4b18-9b17-52f1aa3d02fc}} is only an {{formula:2ea9afb8-8dc1-4d02-b03e-a0601236c208}} valued for {{formula:2ede8350-3a84-4487-b95b-e152f99ab286}} . To have a global covariant topological current, the scalar field {{formula:50cc665c-ef3f-4a07-b392-f33484b02feb}} has to be an {{formula:1a3298d5-2b7c-4366-8a7d-0b7edea24380}} valued for {{formula:25edd0a2-afb3-4cec-a5fb-1d5cd42c4853}} which makes the theory even more complicated because more terms arise as the dimension increases {{cite:5f079fcb613f9ac12595d4f5fb99a2f4d66892cd}}. To see the second consequence, we have to employ the analysis on the (event) horizon and the outer boundary, namely, the asymptotic region for {{formula:a70867b7-ee90-416f-b8e3-6da688a736f9}} and near the cosmological horizon for {{formula:f06d0672-1a50-4917-bb1e-bc8a87cd4bfc}} . Near the (event) horizon, in order to have a consistent picture, all metric functions and the profile function {{formula:28d0a968-67c6-4b54-ae2d-1eba9c742371}} should be positive constants and they are related to the cosmological constants {{formula:0220858c-d64c-4343-8052-76b585f02111}} via an inequality coming from the consistency condition of Ricci scalar such that near the horizon the 4-spacetime becomes {{formula:81b50dc9-5776-4208-a57a-cbee70792524}} where the 2-surface {{formula:ccbde87e-2650-4f97-9ad9-3e92fb6e58b8}} could be either a flat Minkowski surface {{formula:1b6d0ff9-2c57-43a1-ad8a-736c1520338c}} or an anti-de Sitter surface {{formula:8dcaf08c-cd82-4c11-aa7f-ec9e4a3ba628}} {{cite:e465a8c96d1892595eaa16f2893f0afba69a9a4d}}. A primary branch contains black holes with two possible topologies of {{formula:68e27a85-8b01-4150-8131-d2dd5044b201}} , either {{formula:ea6d4c5d-cd64-4013-9211-6a8d89d37259}} or {{formula:2582ec95-fb33-493a-bff6-b3e19202cc5c}} . Another branch, that we call secondary, contains black holes with only one possible topology of {{formula:52aa4927-94ee-4f71-b0fc-f1c8c5b31c5a}} . In this setup, one can immediately see that this classification is simply determined by the value of the cosmological constant {{formula:8625f1c9-06d9-4688-afeb-b3211b25ebf6}} . On the outer boundary, for {{formula:1c1043f2-05e4-4f4e-964f-1e94edf3521c}} case if all the metric functions {{formula:1f48e336-ae1d-4c65-abec-799e9f82ae04}} , and {{formula:836cc160-c51a-4322-adb7-7ebcbfc7e29c}} converge to constants, then the black hole spacetime becomes Einstein geometry. We also find that using the Komar integral {{cite:f35b900d555afc25a3017177af418abcf419d427}}, {{cite:48759503a91ed5f80c32896fce311bb2f3c6e827}}, {{cite:7e569410da26bc3572e267a1270588c50c4e1351}} the black hole mass is generally constant.
| i | ff61019d86b239e6c203d2de2c5d5155 |
We additionally compare our method to Wang et al. {{cite:c9c05cf15b941b1c9193ac4ba5f5b11c2c332d74}} who proposed a recent method for multi-person pose forecasting achieving state-of-the-art results on several datasets, as well as other recent methods including HRI {{cite:dc69880c581407dade5281577bebca0c043103a5}} and LTD {{cite:aa40317fa463d3bbd017141aea3321239cd8efc9}}. Using their protocols, models are trained on a synthesized dataset mixing sampled motions from CMU-Mocap to create 3-person scenes, and evaluated on both CMU-Mocap and MuPoTS-3D. We give 15 frames (1000 ms) of history as input to predict the next 45 frames (3000 ms), and report the MPJPE at 1, 2, and 3 seconds in the future. For fair comparison, we train and evaluate each method using the code and data provided by {{cite:c9c05cf15b941b1c9193ac4ba5f5b11c2c332d74}}. We report in Table REF a comparison of each method on the testing datasets. Our model consistently outperforms the others on both CMU-Mocap and MuPoTS-3D. We observe a significant difference in performance on MuPoTS-3D, which unlike CMU-Mocap is not synthesized from a combination of samples poses, meaning that the data represents real social interactions. While {{cite:c9c05cf15b941b1c9193ac4ba5f5b11c2c332d74}} predict the future pose forecast for just one person at a time, SoMoFormer is able to fully leverage attention over joints from all people in a scene to make predictions for all people simultaneously. When data contains interactions between multiple people, as in MuPoTS-3D, our method can better model these interactions to predict the future.
{{figure:2bcbf35b-d4fe-4513-83d1-6bbccf5e9aa5}} | r | 582d9face6c1d2f376269010b07882b3 |
To simplify notations we shall derive the spectral method in the classical Fourier-Galerkin setting introduced in {{cite:112e13a9e7afa3f60a8863cae221b59e971c7cec}}, similarly it can be extended to the representation used in {{cite:8b47718156beb3bb5f4e8c14436b48e953515833}} for the derivation of fast algorithms. Thus, we perform the usual periodization in a bounded domain of the operators {{formula:068841d1-bfa5-48ca-8884-20f949c2d740}} and {{formula:e710c511-21d6-4cc4-8982-af0cb4f80175}} and denote by {{formula:6d16a068-c3f0-4399-8017-53c2f59fe59e}} and {{formula:1a7759a8-7c56-40dc-90c2-3d4cc8c83a9e}} the operators with truncation on the relative velocity on {{formula:81fba805-c3c0-4705-b084-3522d4320f92}} .
| m | 28a848d071bc4856f4dc3356e3793bf1 |
Datasets & Metrics: We evaluate iFS-RCNN on the modified version of the COCO 2014 dataset {{cite:4f086a174d77a5deb4e2cf786f66d1824ff41a7c}} introduced by {{cite:cbd0656fe263a08de67665cd6484fa4b3b193113}} for FSIS and FSOD. Also, we are the first to evaluate iFSOD, FSIS, and iFSIS on a new split of the LVIS dataset {{cite:ecfd06c6682a28779ec8c626ee0d441bde0b29c7}} introduced by {{cite:d41e999c431e9596ae7e82029a68c023a069113c}} for FSOD. We report the common COCO-style evaluation metrics of both object detection and instance segmentation – namely, the average precision (AP) at multiple intersection-over-union (IoU) thresholds ranging from 0.5 to 0.95.
| r | 3a86bf7a73218edb3b25295967b2561a |
Our method also generalize to NeRF models. We use PlenOctrees {{cite:67d031adf904fda256bcbdb1441c01b8912aaab0}} to extract the discrete volumetric representation from learned NeRF networks. For our purpose, we use the NeRF-real360 dataset which contains 2 real-world 360 scenes. Following PlenOctrees {{cite:67d031adf904fda256bcbdb1441c01b8912aaab0}}, we use the modified NeRF models where Spherical Harmonics are used to represent color rather than RGB values. We then convert the learned NeRF network into a {{formula:786b5c5b-fbdc-455f-a523-456c7347a6b0}} volume following the suggested PlenOctrees settings {{cite:67d031adf904fda256bcbdb1441c01b8912aaab0}}.
We present the qualitative results in Fig. REF where our method correctly localizes and segments the foreground object in the scene.
| r | 99aa09fb05fcae217397f7ea78254583 |
Solutions of (REF ) in the whole space and evolving from initial data {{formula:a5097ddb-8e04-462b-9af4-d0b31d2e1136}} become bounded instantaneously. This is captured by the famous Nash estimate {{cite:8164fd7b470995bde518fafdeb090491e58d0e38}}:
{{formula:1bd49957-3f54-4fad-be05-5cb3e9d7dc12}}
| d | 3c68b61ff3056fd1dbbec7136619ae94 |
We present the numerical solutions to the BEs as shown
in ıineq:BEN,eq:BEL() for the benchmark model defined by the
rescaled Yukawa matrix presented in ıineq:rescaledyukawa(REF ), and for
a tri-resonant singlet neutrino spectrum. In the following analysis,
we restrict ourselves to heavy neutrino masses above {{formula:9c4e4584-fd1d-46f4-afa4-6ec80b6e2d54}} . Below
this mass scale, our approach is more limited due to the fact that by
ignoring thermal masses, we do not account for phase space suppression
effects and their impact on the leptonic asymmetries. Besides these
thermal effects, a more detailed
treatment {{cite:4c54a392bf21809e4c98980770b85eabb85555c6}}, {{cite:47d3add7f9d0b69f587ce83271920c2786e7ee17}}, {{cite:283595679b8dc669d808f650d05f1f1dcdddceca}}, {{cite:2b735e56464d618208ed68e8027365b90074b9bd}}
would require us to incorporate additional CP violating effects
induced by the coherent oscillation of heavy
neutrinos {{cite:f149e2ebf61efb34e5cb28be7f8f1fe4f06506b2}}, {{cite:f0ad3fb4df7b460899f566b501bb4de04fe4889c}}, {{cite:3e73576f6a470a9e70bf49278e4aba01b61658e4}}, {{cite:0531ff86c90f82a96eca7b1a1c28ad68c94d3fe3}}, {{cite:985bf939f1b47853b82cbe1fa915bf90ccfb9278}}, {{cite:d3e6342e5db6655589a34b253c1b5e4d426cb5c2}}, along with those
effects that come from their CP-violating
decays {{cite:2426ef6f5086be4e23ed9e5f9550dbf268c1fd50}}, {{cite:fcb9d59ae56d8be1fbb00a1f0f1d43540ca03d12}}, {{cite:f1c63b884f85517d25abf7d4cebc1af035528911}}. Finally, we note that
for low singlet neutrino masses, the necessary CP violation could
originate from Higgs decays into a singlet neutrino and a lepton
doublet when the thermal effects of the plasma are
considered {{cite:13bef7c2f76146c0343997ab7843672059778046}}, {{cite:af91fe4b08f739bef98e24dc57787a5796db8906}}.
| r | 8d973765a15880801d04b7348a81a384 |
We compare with a recent bayesian continual learing method, CN-DPM {{cite:6850aec5dd1660c6e4783133618473d1d84784b8}}, for completeness of our work. We report the results in Tab REF . As shown in the table, CN-DPM performs better than MERLIN on Split-MNIST, but drastically fails on harder datasets. We note that the baseline methods considered in this work also perform better than CN-DPM on non-MNIST datasets.
| m | 07d997b7d7603f52985d4891761fee18 |
It should be noted, however, that our findings need to be considered in context. We evaluated one particular type of CSA – a chat/text based CSA like those proposed in {{cite:da917d746ab1dca40341c163857a522e9c911b83}}, , {{cite:f339623b3c4e0a4b9aa8579cdeb47933ee69add0}} – where we employed the traditional IR evaluation approach in a conversational setting. We also used simulation based methodology so that we could begin to explore the large evaluation space (which would be near impossible to do so within a user study). Even so, we could only explore a subset of possibilities and focused on pure strategies with fixed rounds of feedback, etc.. Nonetheless, by evaluating and comparing pure strategies combined with the different mixed initiative approaches, we were still able to observe the strengths and weaknesses of the combinations and better understand the different trade-offs.
In practice, however, it is clear that a mixture of different strategies and approaches will be employed and required to optimize the rate of the gain experienced during a CS session. As more interaction data becomes available from deployed CSAs it will be possible to instantiate more nuanced interaction models, and to evaluate other conversational search settings where the costs and gains vary. Clearly, this would change the pay-off dynamics associates with the different conversational turns – and so evaluating different types of CSAs that, for example, try to surface relevant information directly would invariably lead to different strategies evolving. We have also made an assumption that CS should be as efficient as possible (following Grice's maxims of conversation {{cite:25c977c012dae3f9dc56b586455876d3d3b501b4}}) and that the users of CSAs and the CSAs will adapt/evolve to maximise the rate of gain (as per Information Foraging Theory {{cite:96c4f30d0b3dbfaf3ca884942deb5dccda13378d}}). However, it is possible that the conversation itself has additional benefits leading to greater user satisfaction which may not be captured by focusing solely on gain, cost or rate measures. For example, in previous work, they found that asking a relevant clarification increased user satisfaction in voice-only conversations {{cite:34f62478f3f50b2f3d8a9697b418064e67b59dac}} and so this may lead to other trade-offs emerging with satisfaction. Also, in this work, we solely relied on the TREC assessments. In a more realistic experimental setup, one could compute gain based on the amount of useful information given the agent's response. But, these are emerging challenges within the context of CS that need to be addressed through the development of more fine grained test collections before we can evaluate such scenarios.
| d | f09318bb921988d1eed73743a61eafc7 |
First, the SIF method which had been shown to be a robust, flexible, easy to use and yet competitive state-of-the-art baseline by {{cite:a7bcdb56f24d90ed09862b3882b7440ad07055b6}}, has also been confirmed for our technical and scientific domain. Unsupervised general language word embeddings, while not optimized for the use within a specific scientific and technical domain, still proved serviceable in predicting the similarity of highly technical concepts. Even though some reservations due to comparative modesty in size and the small number of raters (however expert they may be) seem appropriate, the strong correlation between predicted similarity and expert judgements can be regarded as proof of concept.
| d | a30e1e19e665474d2a534b1314b4b079 |
Once the isofrequency contours of the infinitely periodic structure are obtained, these can be used to infer the behaviour of a finite laminate, given a large enough number of bilayers are taken{{cite:8be28fe0ebc5e530922bd05997c0e4f4ca9e2dcf}}. The design of the corrugation for a particular effect then rests on interpreting these contours, as we do below.
| m | 1f4176602475bf30d9630bbe207fc661 |
where {{formula:8214597b-29bd-4234-a057-ebed93874b5b}} is the {{formula:00d437ac-c628-472c-821d-62c2e31e7fc7}} 'th k-quantile of {{formula:fcd9aca0-666f-4ee1-9cf9-e2666aaf9cca}} and {{formula:aab289e6-d697-482a-9ed1-bbdb79f5c15b}} As a consequence of Proposition REF , by conditioning on the estimated propensity scores via stratification {{cite:f71dee4e5278b9e302c546a57341b329637c6000}}, {{cite:f3708dd724797888edc563e8a53039e86fa27747}}, we have
{{formula:efff9f26-c07f-488b-a94b-83b8c466b481}}
| m | 6198d5ccac306180912ab866e9b4226e |
The presence of very young stellar populations in the environment is often interpreted as the SNe having very massive progenitors with similarly young ages (e.g. WR stars). However, it should be cautioned that the SN progenitors may actually be much older and reside just in chance alignment. Also note that star formation bursts on small scales may appear correlated if they are controlled by physical processes on larger scales (e.g. spiral density waves that sweep up and compress the up-stream low-density gas and leave behind a trail of sequential star formation at the downstream, {{cite:b89312c8834d8ebd67f096a5888bcbdb7158f2e8}}; the feedback from massive stars which can trigger new episodes of star formation in their vicinity, {{cite:171ae200433adc1fb9c44a21a46a1ab9520edd98}}; and the ubiquitous supersonic turbulence that drives hierarchical patterns in the spatial distribution of star formation, {{cite:956dae24c3787e2c7207746960ddf8831dc70c05}}, {{cite:dc115cadd83f7bbb473a8a916bd916c722ab6a76}}, {{cite:a719a072ddf014d469362f67554bd7d5491fbfa1}}, {{cite:b9663cb38d48808a560d6d5299bb510758c01e3c}}). The relationship between a SN progenitor and the youngest stellar population in its environment may, therefore, be very complicated: the progenitor may (1) be a member of the youngest population and have the same age, (2) form at an earlier epoch of star formation unrelated to the youngest population and aligned just by chance, or (3) arise from an older stellar population in the vicinity of the youngest population, whose formation is related to each other.
| r | d36c0da0988a9e8b77209b0188361115 |
We analyze the I{{formula:b3efc52d-0881-4fe5-b73c-16bb47146cb9}} CDM and {{formula:d8684767-e3fa-41dd-97d1-c6a716c915f7}} I{{formula:41e68b2d-76ba-482e-a8b4-1daf3b92bea0}} CDM models in contrast with their respective counterparts {{formula:58166ff4-c421-4f1b-8960-a7c19b429ad3}} CDM and {{formula:4cd6c92b-1ec7-4b60-bb88-5dfef7bc6d97}} CDM. We also present the analyses of {{formula:369281e7-3247-4ef8-a479-cff079d3f405}} CDM and {{formula:c8a3b886-e3f9-4a09-8a7e-38bc95d979ca}} CDM models in contrast with the I{{formula:c1b1d6c4-417f-4111-83d4-6751215f3c33}} CDM and {{formula:5a1cedb2-f247-47a3-aaff-1e0410fe09a8}} I{{formula:320ebee6-5b6c-4d97-8652-d8026bb8b484}} CDM models. In our analyses, we use the following data sets.
(i) Planck: Planck-2018 {{cite:d3eb5d863225345b7022fad606e2134825658248}} CMB temperature and polarization data comprising of the low-{{formula:29a01901-bc47-4eaf-ac1d-7f9f7e6c1aa2}} temperature and polarization likelihoods at {{formula:c04cff95-5908-49fa-bbaf-88cce3339d46}} , temperature (TT) at {{formula:1aed9781-0ea4-43d5-9a63-1032c6873806}} , polarization (EE) power spectra, and cross correlation of temperature and polarization (TE), also including the Planck-2018 CMB lensing power spectrum likelihood {{cite:9f76acfdebf41c72770ecc76c0300493697ba7ac}}, (ii) BAO: the latest BAO measurements from SDSS collaboration compiled in Table 3 in {{cite:5a028c4877ba400263df0f1aebdbf796dd50a221}} (iii) JLA: the compilation of Joint Light-curve Analysis (JLA) supernova Ia data {{cite:9901d4d6677c79f1396fe9056077ab461670e64a}} (iv) KiDS: the measurements
of the weak gravitational lensing shear power spectrum
from the Kilo Degree Survey {{cite:43a28c6434fa7f777c889fafaf21aba2374f53fa}}, and (iv) R19: the new local value of Hubble constant {{formula:754cfd09-7074-4561-ba69-bcf9a1e2b6c3}} km s{{formula:38e1b820-3fcf-403f-b584-1a6802c22bdd}} Mpc{{formula:e6ca908b-320f-4a56-9e65-382043a17094}} {{cite:b457202d97d556259cc61978d7f6826996a79a6d}} from HST. First, we analyze the I{{formula:045f9d84-665d-40e0-a580-9823dfb3dc92}} CDM and {{formula:a1d991dc-08b0-45f7-9b04-50fb16a12b9a}} I{{formula:9c491112-813a-4622-9836-0cd22255bf16}} CDM models with Planck+R19 data in order to explore the features of interaction in the dark sector. Then we assess these models by considering the combined data set: Planck+BAO+JLA+KiDS+R19. We present these analyses in a comprehensive way while doing a comparative analysis of the six models under consideration: I{{formula:f77297a8-4550-45e4-8d95-80597b139736}} CDM, {{formula:8f805018-3aab-40e0-aa5a-e0215881ad10}} I{{formula:9da091db-076c-405a-8099-5f4d68f8bff5}} CDM, {{formula:a9396ec8-8e24-4e70-8957-344b66d727cf}} CDM, {{formula:2ca46b31-9457-4eb2-9d8f-665a76852e30}} CDM, {{formula:0a2864c5-88c6-4f91-8f6d-fc322213ecba}} CDM and {{formula:f990fef2-cca2-4260-86e5-56f7ceee333a}} CDM.
| d | 232b579bc91a43a6a9202b6495e2cf88 |
The interpretation of data from stellar observations requires a stellar model to compare against. Traditionally, these models have been 1D global models {{cite:11167d469f65f6ee2f49c06385a488497acd72b6}} that use formulations of mixing-length theory (MLT) {{cite:77274f4e0cf0c96e93849825a45cd7f12619dbe4}}. Later models accounted for line-blanketing effects and used an opacity distribution function (ODF) approach to calculate opacities {{cite:62df861869cef3c36be8a647ad76f3e74459882e}}. Among others, the MARCS code {{cite:287f68250b12f56f1e9e6a27d98f3cec31c3607b}}, {{cite:9893b867a54a260bcc981743765c0abe09418251}}, the ATLAS code {{cite:b5f15af76fb7c2013efccd2e1cc8dc434052dd6f}}, {{cite:7f9c4c971a8086e5d2940e5d8515e25dde4f8ff6}} and the PHOENIX code {{cite:6d614b47f79e183ee3c436a2b4edd239e0f7883f}}, {{cite:e3ff7e4350ed808fc29fd42589ec402e51ffe037}} have enabled the calculation of synthetic stellar spectra with a detailed accounting for the relevant physics. These models enabled, for example, the accurate determination of abundances, stellar evolution tracks, and even for constraining the chemical evolution of galaxies {{cite:5c368e197f6b8d6753625a9604bce03ed9799f27}}.
| i | 96ed55ae8aa1c1b395c52aba09cff28b |
One research line is to formulate the drug generation problem as a sequence generation problem.
Most of these methods are based on the simplified molecular-input line-entry system (SMILES), a line notation describing the molecular structure using short ASCII strings {{cite:42f59a2b9bf1ce47b0a5f24d08b35e745deebaba}}.
| m | b51844002604ccc8dca73d6457b3d7a1 |
In relation to the neutron star properties and the astrophysical constraint, one of the most robust constraint is the minimum mass that a EoS must reproduce due to observational measures of massive stars via Shapiro delay.
Not long ago, the neutron star king was the PSR J0348+0432, with a mass of 2.01 {{formula:0b0f711a-4544-45bc-b22c-1ef6926ec8a7}} {{formula:b659e35c-d9d8-482e-9dcf-35af4a279b37}} {{cite:0492861a38de4580511a2facf563335e86b19007}}. The old king was deposed, and now the PSR J0740+6620 with a mass of approximately of 2.07 {{formula:63d73e68-3142-434f-a9ac-6f58223ea69a}} 0.07 {{formula:712f60b5-97aa-44ef-9e8b-355612fa797e}} {{cite:ee97ff5bace480893b91187507e8a79813ce1551}}
rules over the sky. Notwithstanding, both pulsars send the same message: the EoS must be stiff enough to produce a two solar masses neutron star.
On the other hand, one of the most controversial contraint is the radius of the canonical 1.4{{formula:ab6fe979-5d4c-478d-86c7-7dd3659f99b9}} . Just in the last couple of years, different studies point to different values for its maximum allowed value. For instance,
a maximum value of 13.8 km, 14.2 km, 13.5 km, 12.6 km, 13.5 km, 12.8 km, 13.0 km, and 13.2 km was found
respectively in ref. {{cite:7f169705421aebf7c807722f423da37d2c670bd8}}, {{cite:27163695c55a75b20a6c461b27586055824258a6}}, {{cite:1ab43fd8a341739e8fbffbb0655bc9b3fb787730}}, {{cite:136453bdf43bbeddde2fb59696050eb14a94081a}}, {{cite:0a0424e4720a52cf8c5c6cbfb4e21a30382db6a6}}, {{cite:3f5c7c2ff9824a19eedb062a446eb1db70779730}}, {{cite:55389264cb8a659b38cd51b4ca99c5092ac3eed1}}, {{cite:f0d57c3e1a8465d17561297dde7bd5a5a318c5be}}.
The average value of these studies is 13.32 km, and I use it as a constraint.
| r | c2b06631d7960903287d69c38167dafd |
The latter can be done using existing interval methods that compute interval enclosures
for the united solution sets to interval linear systems, see {{cite:78dd339a0d6cecc51f0f99074636682e2a0acc37}}, {{cite:a244686589d3af403f0df2daf56aa18ca8a3f1ee}}, {{cite:f27fdcb1cf1dac137fdcd341c5076d1d0daaba5a}}, {{cite:124e2c759158ad6f611b02d30e32d1b19f2cc950}}, {{cite:192496576e34fba33e852eba02963fe77139d66f}}, {{cite:aac697da7e5d8f5cbfb7a30d2ba46db12e099631}}. If, for any interval
linear system {{formula:155f0dc8-4cc3-405e-b836-c538182cd046}} , we know an enclosure {{formula:a0aa6ccc-6e68-4bcc-8195-15e8c25a5e98}} for its solution set,
{{formula:21f12315-ee3b-416c-be9f-de5d3f7efca7}} , produced by an interval method, then
{{formula:02fa6d7c-1416-4d0d-a6db-ec06ae08f99a}}
| m | a96da082ac30d77a4a6296ed26eb48ae |
The task of Automatic Speech Recognition (ASR) involves building systems that can transcribe spoken utterances in isolation. One major problem with building efficient ASR systems is that they are data-hungry {{cite:6a3bded89860f6779b89de5eb404a5ab2ccdda6b}} and with the introduction of deep learning to build ASR systems, this problem has amplified. Though English has more than 100k hours of human transcribed data freely available online, for languages beyond English, data is scarce, with some languages even lacking professional annotators and existing only in spoken form. Transcribing data is both an expensive and time-consuming affair. With researchers and businesses finding the true potential of ASR systems in various Natural Language Understanding (NLU) systems, there has been an increasing demand for building such systems in languages beyond English.
| i | b25a8d3f7b0ddd3649e1c60e6ab2a949 |
Ethical considerations. Since GST generates the visually-grounded dialogs, our proposed models have the potential to produce biased and offensive language, although arguably to a lesser extent than the open-domain dialog {{cite:e56309519f02307c1743fc7afa54e60d5b666602}}, {{cite:f2b8bb88fa98bdb24a2acc3fba583dbec5fc19c6}}, {{cite:a59d407080bc90339abf6a0c93a363d96ddc378b}}, {{cite:bd2ef20ca550438637282c3d7bc35861e08b4851}}, {{cite:459f96dba35373789125d5f38a2e9d2e8201924e}}, {{cite:8f2c6c1a2edc033ed02cbccff350869c42f4fe43}}. We attempt to mitigate ethical concerns such as biases against people of a certain gender, race, age, and ethnicity or the use of offensive content. Our proposed method utilizes the images and the captions in the Conceptual 12M dataset {{cite:d9eacd48b1d01e17773a719c9262c7cc52cca1d7}}, where several data cleansing processes (e.g., the offensive content filtering or replacing each person name with the special <PERSON> token) have been conducted. At least, we could not find any conversation violating the ethical considerations in a manual inspection by visualizing {{formula:daba631c-f1a2-4cb7-b2d2-3a527e4bd623}} 100 synthetic dialogs.
| d | 27c6e323db110ca56936df2b93e892be |
We begin by briefly summarizing the mathematical results we make use of in the paper. For a full discussion, of the Resolvent/Green's function method in RMT we urge the reader to consult one of the many excellent reviews or textbooks in the field {{cite:f7c381d9d2514365ff77812b969a0284335c9371}}, {{cite:bb31fd86afb66e3292eef5dab5f1588c488dd20e}}, {{cite:90271464980d8ef04f8d30809ff4cdd1f7c27b21}}, {{cite:943ebf1928b0bf97562f080e6e8de09dfa068cd8}}, {{cite:3b1c0553ae3983eb3d6f722a036fdcc2d8179608}}, {{cite:f489a59b0c47a26465e6caf18da4c200b16a1121}}.
| m | fb4946bdd38bff7d4845d6873b93b875 |
In the same way as a number of projection methods of feature extraction (e.g.
Projection Pursuit {{cite:1d5ab68e902816d7e8567ca4c2b0fa7c5f3cae27}}, Partial Least Square Regression
{{cite:43a17c0ea3a44a731ba2d95ea78da6a422b806a1}}, {{cite:b8883910636576a8178f35cf9963c82b8742980b}}, Conditional Minimum Average Variance Estimation
{{cite:684689e5c7dc01ad83f87b39de177c7079147910}} or Sliced Inverse Regression {{cite:979844ee127589d5ea34ce1a1797892ba504a207}}, {{cite:51de55882067c89bc60f0aecaecdbbd2888ec60b}}, {{cite:3e6443f43da8adf73ae50d2f7d042a0e94220700}}),
when implementing the NGCA we decompose the problem of dimension reduction
into two tasks: the first one is to extract from data a set of vectors which
are close to the target space {{formula:5f7c8b72-cc3d-4ba5-a171-5af2d5e0dd04}} ; the second is to construct a basis of
the target space from these vectors. These characteristics can also be found
in the unsupervised, data driven approach of SNGCA, presented in this article.
When compared to available dimension reduction methods (e.g. Principal Component
Analysis {{cite:4fede936693ee733620952c9c3b95008fd6d2d74}}, Independent Component Analysis {{cite:dfa75b04a89efe5156707cebb2e3dd3aa625e189}} or
Singular Spectrum Analysis {{cite:d3c8b52225fb11f5358bce034b33a740361070eb}}) SNGCA does not assume
any a priori knowledge about the density of the original data.
| i | 36ad177b062b6e3142c27533f6e734d0 |
A variety of models have also extended beyond continuous/binary models for the mediator and outcome. These include models for zero-inflated count, survival, and ordinal data, as well as quantile regression models {{cite:acf2aa3f9c564c5e1d3cef559d0d9db02aa47d10}}, {{cite:09220b1bc6b114ef91b432cf4b14fb5f68593f56}}, {{cite:e16c9e876f4698d6a575b4527b44610ff4498da7}}, {{cite:537846fe9d42a6aa5c2ccf35f8f721342c4209fc}}.
| m | 2661bbf8567802cf52768e6f7b543f0f |
with Dirac delta initial data {{formula:47f31232-981d-4076-a432-a6dada0785be}} Here {{formula:c69f4314-de99-4447-bda2-e27a92f70896}} is the space-time white noise. The SHE itself enjoys a well-developed solution theory based on Itô integral and chaos expansion {{cite:704888af46f2de420ae2c42209579ce3b67eaa86}}, {{cite:550912da9b729f07b22508cf47205b7f8cb80a24}} also {{cite:3249b13f2cb33b3c05649648fbc9fc65b6d95daa}}, {{cite:88065f98d7f30cc5158686ef7c97b77e21b5132c}}. Via the Feynmann-Kac formula ({{cite:d4df4e6c0102b451a133c9d0b5a00c6e459ba2c3}}, {{cite:d822aedb8f39708c6e6257945c50674661552539}}) the four-parameter random field can be written in terms of chaos expansion as
{{formula:ed3e620c-4775-40b8-b605-e6cef3fe85a0}}
| r | 5843c698f7d425e3cd74234f437d7529 |
The first method given in Ref. {{cite:43d0162340092dce230ee136744fc36349eea399}} is based on semidefinite programming (SDP) {{cite:809a6b0f417c46c077a8a352fc73a087e4b5110b}}. One starts with an antisymmetric state {{formula:f3cdb6aa-55d5-48ca-9ad5-cbbeb5a7284b}} . Then the task is to find the PPT state {{formula:f3c29c81-2555-4b37-8dce-16f881e9d67f}} (a positive semidefinite matrix of trace one, whose partial transpose is also positive semidefinite) whose projection to the antisymmetric subspace is proportional to the starting state (that is {{formula:2ec278a8-cfec-44df-91ff-3a3d3018027b}} ) and whose overlap {{formula:999fd08b-6a81-4eb2-9924-9f836147368c}} with the starting state is maximal. The procedure is successful if this maximal overlap denoted by {{formula:dacbfa31-ae10-43c5-bc9e-efbc41ca98a9}} is smaller than {{formula:c7f083d5-f379-41bb-8674-0b022ba1033e}} , as it is proven in the paper {{cite:43d0162340092dce230ee136744fc36349eea399}} that in this case the state {{formula:51b544ae-20d9-419e-b2c0-f3811a3e300f}} is entangled. If the result is {{formula:0ab9e2b3-9a09-4765-9cba-d758f7dff72b}} , which is the upper bound for this quantity, one can start from another {{formula:ef7ccd57-d4a7-409c-94ec-f056cd6f0471}} . It has also been proven {{cite:43d0162340092dce230ee136744fc36349eea399}} that if {{formula:5d1decf1-4dcd-4d0a-a504-f3f270b78a59}} is an optimal solution of the problem above, so is {{formula:a8bfc6ba-6e1f-47b5-827f-1643bb63ded7}} , which is a convex mixture of the starting antisymmetric state and a symmetric state ({{formula:d2d6107b-1936-4ff1-b96a-d15b7e625a1d}} properly normalized). Therefore, an alternative way to get {{formula:551925fb-dded-4b7c-91e1-26b335cb862e}} is to find the optimal symmetric state whose convex mixture with {{formula:881c241d-a9c2-4d24-96ad-abfbe9d095f2}} is PPT, and the weight of {{formula:5088e0ad-3779-4530-b518-0f3d532320c9}} in this mixed state is maximal. Then if this weight is less than {{formula:6c389539-e2c4-4580-8525-6c205e6ba3b5}} , the mixture is entangled.
| m | f5f10c87cb76624631ca4a0c4a2572ed |
To close this work, let us give some comments about this project.
Due to the axion, the quark potential and entanglement entropy is
shifted as some holographic study in four-dimensional QCD with an
axion e.g. {{cite:05a701b5da58d18be307406b0eca1600ea65e7d2}}, {{cite:d836637f10e7fcf6edbe3b9775c4d28adcc1575b}}. However our work additionally implies
the quark tension and the potential phase transition illustrated in
the behavior of the entanglement entropy could be destroyed in the
presence of strong anisotropy. These can be found in Section 3.2 and
Section 3.3: when the anisotropy increases, we can see the quark tension
trends to become vanished and there would not be a critical value
of {{formula:fdea1976-3913-4cfd-9a70-c29ff3660c64}} satisfying the entanglement entropy {{formula:771cba59-77bf-48b6-8d2f-90314d1d4f14}} .
As the entanglement entropy could be a tool to characterize the confinement
{{cite:cca97625841a8299aef4758b4d6f178633097afd}}, {{cite:d3f6553922c0727aad91f17e3d8b42303ab5e3d7}}, {{cite:6633759d11e5777989f8ca9fc2a31255d6a26224}}, {{cite:d8cd35bb4eb9115256f5999d5f2fa0af6c8500dd}}, this behavior implies there would
be no phase transition for {{formula:3ff9fa42-a630-467e-8642-b835a56100d5}} i.e. no confinement for strong
anisotropy. Besides, the baryon vertex also reveals the unwrapped
trend when the anisotropy becomes large and the “unwrapped baryon
vertex” also means deconfinement {{cite:762340188f14b826b9758c2c11e4cd313e0c423b}}. In a word, this
holographic approach shows us the confinement can not maintain in
an extremely anisotropic situation. Interestingly, this conclusion
is in agreement with the fact that the QGP is anisotropic and deconfined,
so it may provide a holographic way to understand the features of
the strong coupled matter.
| d | 4c2531a05afaf4db2132e0a1d8a6459f |
Densification of wireless network is a promising future direction to satisfy the explosive mobile data traffic demand.
In response to an increasing demand for data traffic along with the use of mmWave and terahertz bands, ultra-dense network (UDN) where a large number of small cells are densely deployed on top of macro cells has received special attention in recent years.
Since there are many small cells close to the mobile user, UDN reduces the path loss and also improves the quality of line-of-sight (LOS) transmission {{cite:e093a310045722eb1efe8bb7cbd2d7e5b9283005}}.
Furthermore, UDN expedites the reuse of spectrum per unit area dramatically, resulting in a significant improvement in the throughput of the wireless network {{cite:af11a30fe4569cf9fe2316154e865aecff9eead5}}.
However, energy consumption of UDN, caused by the exponential growth of a mobile data traffic, is a serious concern for the network operation {{cite:562eb3173e834e697bfe1936e1f348e016d44ea7}}.
In fact, an upsurge of energy consumption is a heavy burden in the operational expense (OPEX) for the network operators, not to mention the increased carbon emission and the acceleration of global warming {{cite:78ba3e03aa726f23f316fabeb1a63853ffccc37a}}.
Pursuing the balanced efficiency in energy consumption and throughput of UDN is, therefore, an important direction to ensure the sustainability of the next generation wireless communications {{cite:c406085151ecb4610d24eb0d493c9449e4fd6b66}}.
| i | de6bb18002d48ed68151a1cc55b5f462 |
Recall that for arbitrary matrices {{formula:c109a386-cc5d-407f-a750-e09d2aeabc24}} and {{formula:a70e6bd6-d65d-433a-aabb-3160c58db9e9}} it holds {{formula:3e9a5b6e-854c-4f4f-b625-d6e0204a93cd}} {{cite:dc9c2dc8434443633512dc4a01a95098f9124329}}. This together with the Hölder inequality with (3/4 + 1/4 = 1) and Lemma REF with {{formula:8103a307-a568-44a1-8db3-a0b01685b5ad}} gives
{{formula:083907aa-2b10-4c00-ae50-1042ad23c47f}}
| r | c7c25653f5994e93eb85e23d02d7b308 |
More practical approaches in this regard have involved novel changes to the optimization procedure itself. These include adding a “momentum" term to the update rule {{cite:6decd7ff659ec8f6f11dbc3f4fff86afbd7e7a46}}, and “adaptive gradient" methods such as RMSProp{{cite:3ac8f9bbb2ae004e6ad87dc79457df039a24ef85}}, and Adam{{cite:088597cab64581912efcb271dc33676f8cc01ea8}}. These methods have seen widespread use in deep neural networks{{cite:62e3644d0b0460c7ab582f25d7cb37add989c73a}} {{cite:6829c6cc7e406bf40dc0e0222a6a2a4efeeb3c98}} {{cite:fa5e22974376ee41bf015ab1de22e2977a4ef524}}. Other methods rely on an approximation of the Hessian. These include the Broyden-Fletcher-Goldfarb-Shanno (BFGS) {{cite:6882a805c17e01b44ae3d7c7a1a251dded456420}} {{cite:86dad93f2b9ddda85f876cd6c219b0b34a72eb93}} {{cite:24a27ec09fad53e20fdd48781d3e7e079cddf710}} {{cite:4da68eb04c4bdede0a7cb9f383b9945059e65de3}} and L-BFGS{{cite:9719b79e579818f11b5ee77da62a4d7ff9c0ddba}} algorithms. However, our proposed method does not require any modification of the standard gradient descent update rule, and only schedules the learning rate. Furthermore, for classical machine learning models, this learning rate is fixed and thus, our approach does not take any extra time. In addition, we only use the first gradient, thus requiring functions to be only once differentiable and {{formula:dab5b649-3cb0-41ab-b01f-c70a85907a8d}} -Lipschitz.
| i | 4486600b0a568e4f238b04b812e51c16 |
where {{formula:4f88236b-adfc-4709-9fc6-501fc693d549}} and {{formula:50d6e002-5ccf-4604-994c-aa44d97eb6b8}} for all {{formula:57c86e96-4050-43a7-9513-2545335cb1cb}} and {{formula:e482b8d8-d798-4b38-8a80-e8746eebddfb}} . Since under Assumption REF {{formula:cc36caa9-a8cf-447e-8fd1-c092bced36be}} is assumed to be strongly convex, there exists a unique global solution {{formula:0022ce29-0f2e-47f8-84c1-0296efa287dd}} of (REF ) {{cite:d4cf1856ae6bdfee834a8a58a4a44ef4c64d2257}}. By the Lagrange multiplier theorem {{cite:d4cf1856ae6bdfee834a8a58a4a44ef4c64d2257}} there exist {{formula:0a418a25-a7bf-46ee-913f-5a9a1cbb4791}}
{{formula:91ddd35f-be30-45d0-95da-38b79a4b3817}}
| m | 84cbbfc7a48bfbdc3dc15b7f37290725 |
Moreover, our method significantly outperforms previous methods on the precision item, which attributes to the false-positive filtering strategy.
In the end-to-end case, our method significantly surpasses the best-reported results {{cite:3e1153ca47216aa4f2c8cb188bfcee29112d13d2}} by 15.7% on `None' and the best of results {{cite:ce3ab7653c804dfdbc156a909b20bbe99bdf0441}} by 6.5% on `Full', which mainly attributes to STM achieving the end-to-end training strategies.
Since CTW1500 releases the recognition annotation recently, there is no reported result on the end-to-end evaluation. Here, we report the end-to-end results lexicon-freely, and believe our method will significantly outperform previous methods.
{{table:bb7f016f-f1fc-448b-843b-d2990f86daa7}} | r | eb684448bc7615eff8c8124c0557ed48 |
The goal of this work is to propose a conceptually minimal combined contrastive masked autoencoder approach, aiming to find better trade-offs between simplicity, efficiency, and performance. Consequently, we choose to omit a number of commonly used self-supervised learning design components. For instance, we do not use a momentum target network or multiple views (multi-crop), since they both increase memory requirements and run time. Even without these commonly used components, our minimal framework achieves very strong performance compared to prior work, and importantly improves performance over its contrastive and autoencoder constituent parts. We expect that a wide range of modifications, such as momentum target networks {{cite:0d388ead09353bdcfe30f5bd367f21c96fbdd9bc}} and multi-crop {{cite:235c8d8c2b821dd7c2efbbefb1eef18b7e5bf750}}, will improve performance further on top of the core method.
| d | e068b69eb21b75b22f8b87d34856591c |
We have applied the LCO formalism to a new loop order for any theory and in
particular a gauge theory. Our main focus has centred on QCD with {{formula:4ee86062-9e9e-48bc-be9a-986999179b49}}
massless quarks. The key observation is that for Yang-Mills theory the three
loop corrections to the effective gluon mass derived from the minimum of the
effective potential demonstrate a degree of convergence. For the {{formula:aa3a9f38-33cb-47c2-8a7f-98024634ef61}} colour
group this is centred roughly around the value of 323 MeV for the effective
mass. This is not inconsistent, for instance, with values from other methods
such as extracting a mass by fitting perturbative gluon propagators to lattice
data or functional renormalization group techniques. The same analysis showed
that this apparent convergence was evident for {{formula:e1c0e467-f2a1-4663-be55-105ab05e43f2}} Yang-Mills theory as
well. This situation for non-zero {{formula:c2f1b281-fb10-45a0-8513-3c669af233c1}} was not as clear cut. This was
primarily because as was noted in {{cite:97a9cbe5c0f0ef12a6169c206e59249f1898d9df}} there was a marked difference
between the one and two loop values of the effective mass computed there albeit
with a different definition from the one used here. With the definition
(REF ) used here this is also apparent but interestingly Tables
REF and REF support the notion that the two and three
loop values for the effective mass have stabilized when further corrections are
included. Of course this is in the simplified scenario of massless quarks which
is not truly realistic. What would be needed is a generalization of the LCO
method to induce quark masses from an extended effective potential. This does
not seem straightforward if one naively examines the core formalism. The LCO
construction for the gluon mass operator benefits from this having mass
dimension 2 with an associated mass dimension 2 source field that therefore
requires a quadratic source term on renormalizability grounds. Using the same
dimensional analysis the quark mass operator would have dimension 3 meaning
its source field would be dimension 1. Therefore aside from the mixing of the
two types of sources to produce a cubic term, quartic quark mass operator
source terms would be necessary on renormalizability grounds. The technical
issues of accommodating this within the LCO formalism may not be insurmountable
but would require new insights to find the effective potential.
| d | 176ee47d5abdfe7d1ffbe487b745a5b7 |
Finsler geometry as a more general geometry could provide new sight on modern physics. It is of great interest for physicists to investigated the violation of Lorentz symmetry {{cite:8b4b3ec4823473b3affcf1efe50be7f5b8dec230}}. An interesting case of Lorentz violation, which was proposed by Cohen and Glashow{{cite:0a96c82eddcdc549206484ea6bb423a090a309eb}}, is the model of Very Special Relativity (VSR) characterized by a reduced symmetry SIM(2). In fact, Gibbons, Gomis and Pope{{cite:593246e5bda312946921e4eada7172197ea632b9}} showed that the Finslerian line element {{formula:9aebd344-f74d-4bf1-b1f3-4954626a6410}} is
invariant under the transformations of the group DISIM{{formula:9ce77038-6cd8-4d5a-a32a-c556f39bfcf0}} . In the framework of Finsler geometry, modified dispersion relation has been discussed{{cite:9911a2814755d09e7a21b5391d48aefa69c6f9f2}}, {{cite:1eb2bf9b6f40f4d1b7eba7b65b8f74fb769e8d52}}. Also, the model based on Finsler geometry could explain the recent astronomical observations which Einstein's gravity could not. A list includes: the flat rotation curves of spiral galaxies can be deduced naturally without invoking dark matter {{cite:e937935ec07bbd4f4b5d0a7ef77852d24f392aa5}}; the anomalous acceleration{{cite:2d2e98af60acf31c692039faf41aca2280cff8b9}} in solar system observed by Pioneer 10 and 11 spacecrafts should correspond to Finsler-Randers space {{cite:47cd8401b0bdde27e3e3286f106930cbb07073be}}; the secular trend in the astronomical unit{{cite:341350812d5168bc9f6dcf63e3a7f330c13d90f0}}, {{cite:4c750d0f7ca2ed3556c19338e7b848371ce8f651}} and the anomalous secular eccentricity variation of the Moon's orbit{{cite:369a70d0d4cb908e6a59704eac95dc3474c56d79}} should be correspond to the effect of the length change of unit circle in Finsler geometry{{cite:fd0a32053e96d26c2029ed3e7f18651f9a92e68f}}.
| i | 0d334dd1a55225d1d62b32ae93f5fc5e |
Limitations and Future Work
There are two potential improvements of GroupViT to explore in the future.
Firstly, GroupViT's performance is lower on PASCAL Context versus PASCAL VOC. This happens due to the presence of background classes, e.g., ground and road in PASCAL Context, which are less likely to be labeled in text; and misclassification of correctly grouped segments into incorrect textual classes (details in supplement Sec. REF ).
Secondly, GroupViT's architecture currently doesn't integrate segmentation-specific enhancements, e.g., dilated convolutions {{cite:ebf0d32e4334f8ac9669684093a7815dcec4dc7b}}, pyramid pooling {{cite:6f248d2659bb45c484493658de8bf091da0351d4}} or a U-Net {{cite:c92ea364a67168a6759ac1f1066ab0029dceab39}}.
| d | 71fd4a229351d5a8286111a45eed93ce |
We recall from Chapter I of {{cite:d5b1659709259d9fce60cc57fc908f3a078a1c6d}}, that a closed subspace {{formula:72d86b3f-07e3-4557-a8b3-5688d144b8a6}} is said to be a {{formula:012d0b89-781c-4f95-8098-3a5ce1058c99}} -ideal, if there is a linear projection {{formula:3a936e68-b4fe-4263-8885-c6d419b50b3f}} such that {{formula:502533af-0420-4420-bd24-90eb746a29c0}} and {{formula:4cadd7c9-8676-4022-ad95-292349d56bfe}} for all {{formula:9af98410-3813-40a2-8b39-52e3ecdb5362}} . When this happens, functionals in {{formula:e17e05f0-161b-4fe4-ab96-c3a045eeb17c}} have unique norm-preserving extension in {{formula:d30822b0-3c80-4e7c-885f-4de90c142fe1}} and {{formula:32094f28-5fde-448b-87b9-381841d3a94a}} is canonically identified with the range of {{formula:dbe1bbdb-0af6-46a3-b604-3cb767f683a2}} so that {{formula:7b04b11d-2d35-4829-952c-5e1166de7a4e}} , consequently {{formula:b9b4ed63-d68a-4439-9dc9-9b0c2edc6bf6}} .
| i | bf3ffb9c49dd605463b50e9b470c2a18 |
In {{cite:3168d05dc42ee11355462d4715542bc600da169e}}, the reverse perspective network is applied for object counting to solve the problem of input image scale variation. The perspective estimator can calculate the perspective parameter and the coordinate transformer can convert the images to similar size. The weight of ground truth is promoted for training by using an adversarial network. The proposed method can obtain the satisfactory result for the dataset with large variation of scale. The frame work of proposed method is shown in Fig. REF .
{{figure:a594bed0-da19-4c7e-90f1-306bab660699}} | m | 22cfe13dd0d620604880454930a64099 |
As far as the non-thermalization of other observables
than the energy density in (REF ) is concerned, obtaining rigorous
statements turns out
to be rather difficult, while non-rigorous
arguments are straightforward and still quite convincing:
For simplicity, let us focus on observables {{formula:725a6051-97ca-4f39-8b77-c5798a94d817}} which satisfy
the ETH (non-thermalization in the absence of ETH is
quite common anyway).
Then, the long-time average of {{formula:cb83c235-d579-49d1-96cb-6c2139bd877a}}
is well-know {{cite:846ed2d45388e389976424e31f9799f25c0c7be8}}, {{cite:34f4ebe4e5b90d91c8d78e5d2c7ffe40e4b45a44}}, {{cite:198ce86a49fddbc037bfea17b21d6b863811720c}}, {{cite:d492650f8185fbd3b2f2e80ca7c2b5059176042b}}
to be closely approximated by
{{formula:dd0e75ee-f397-4ea9-9e46-9e05ee6ebf33}} ,
where {{formula:59c641b5-9803-4f80-a7d0-1a45af7b79b7}}
are the populations of the post-quench
energy levels {{formula:1d126de2-d939-4233-8547-d20c30e59978}} by the initial state
{{formula:4640a120-54dc-4c57-86d8-2ebb5118d2d7}} , {{formula:b565f649-9fc6-4667-94f3-f51f14f71d50}} are the
corresponding post-quench eigenvalues,
and {{formula:3a2284e1-dcd5-4148-89ac-65e28c1803b4}} is a smooth function of its argument {{formula:bdc51f4e-1b70-44a1-a8cd-0b4c5613414e}} .
Likewise, the microcanonical expectation value
{{formula:b4689a06-9e24-4695-8b19-122d5c85f230}} is closely approximated
by
{{formula:3777c4a0-bf14-4391-bd5c-04b91a582ef8}} , where {{formula:97c46daa-3555-440f-a760-ac6f4701119a}} is given by (REF ).
A necessary condition for thermalization is that
the long-time average must be close to the
microcanonical expectation value, i.e.,
the approximation
{{formula:89289e96-5391-41b3-b0b1-b61b197fee32}}
| d | 5faa9447f6eb11aa6d4f4794b65099f9 |
Lemma 1 ({{cite:21ed186f0cc0f6b06da46cbd3b7f29f6b82e99b6}}, {{cite:a8f80336cdfad87b7349df07d89244bc421f6707}}, {{cite:61d7eaeb0293b577491c34c114920bebfbf670af}}, {{cite:205afe9ebeb140970394bc5643201591f9ba5ec3}})
The connection matrix {{formula:b0ff07a0-16f1-4b9f-b990-361f9be97aec}} of {{formula:4ba708d0-5c09-4180-bbd4-71a8922cafcd}} is determined by the following structure equations,
{{formula:24b08324-7e27-48fa-9456-f48085dd5f4d}}
| r | a9eb8f031d90525eaff1a9b66b4ea78a |
The overall pipeline of the proposed model is illustrated in Figure REF . An input image is first fed into a pretrained ImageNet classifier to extract its image feature and a probability distribution {{formula:b8141ba8-01da-4da5-a32b-ed15946466a4}} over the ImageNet classes {{formula:f986faf7-c2df-4d2a-9954-4c525c3ef485}} . We first construct a graph whose nodes are classes from the target dataset.
We calculate the pair-wise Wu-Palmer (WUP) {{cite:46c6fcaac1a27bf367d249b8294b52d0418334f9}} similarities on WordNet {{cite:f7ffd013d7462c703771b2ffac0f2319020ef785}} and then use a threshold to determine whether an edge exists between a pair of nodes. Self-connections are also included. During training, only seen classes {{formula:7559ea40-3ee6-431c-aa92-00009c8dce1f}} are included into the training graph, while during testing both seen and unseen classes ({{formula:8c6a43de-34ae-4993-ae12-0437dafba688}} ) are included in the testing graph. The classes {{formula:b8d8deb2-d9f9-426e-823e-c32198ef3d52}} from ImageNet {{cite:03351649a0996dbc37d1f042a96dba1f07d2d9d7}} are also added to the training and testing graphs, and edges are added in the same way as described above. The overlapping classes between ImageNet and the target dataset have been removed from {{formula:4015958c-1e76-4092-9ec3-1f15d1053847}} .
All aggregated edges form an adjacency matrix {{formula:7d689e8b-e772-420f-8d76-7186fd7fae25}} for training, and an adjacency matrix {{formula:c7359e8a-83ef-4041-956c-48dcc6565fab}} for testing (with little abuse of notation).
| m | 4d3df5a93ce76e8d785dafd6be226a76 |
1) IRTK: One of early image registration tool for breast MRI images using voxelized mutual information similarity and free-form deformation model {{cite:340e8a20e74df1dd4da86cd4a22bd955b8f87374}} . Before starting registration IRTK apply contrast enhancement to make similarity measure insensitive to intensity change. A hierarchical transformation model is applied to capture global and local motion of the volume data where global motion captured by affine model and local motion is by non-linear free form deformation model. Voxel-based Normalized mutual information is used as the similarity measure.
In this evaluation, we followed the same settings as {{cite:9522e916a5ac44814a298d93ab9678494fba2876}} except B-spline control points. Since the pixel spacing of CUBIC dataset is very small, the control point spacing is set to 5mm which is the highest possible value for this method.
| m | 34b982c674a4e6219a6af62a65f96a1b |
Table REF presents results using pre-trained BERT features.
We extracted features from the pooled output of final transformer block as these were shown to be working well for most of the tasks {{cite:cef36d9f67d4ef6db85477899b6b1d764ecc51e9}}.
The features extracted from a pre-trained BERT model without any fine-tuning lead to a sub-par performance.
However, We also notice that ToBERT model exploited the pre-trained BERT features better than RoBERT.
It also converged faster than RoBERT.
Table REF shows results using features extracted after fine-tuning BERT model with our datasets.
Significant improvements can be observed compared to using pre-trained BERT features.
Also, it can be noticed that ToBERT outperforms RoBERT on Fisher and 20newsgroups dataset by 13.63% and 0.81% respectively.
On CSAT, ToBERT performs slightly worse than RoBERT but it is not statistically significant as this dataset is small.
{{table:7be4e68c-8e94-4f1a-bc7b-52650501f810}}{{table:ac7868e6-d915-49d6-aadc-2955aefc8bde}}{{table:7e5a2fa2-bff9-4dc1-adda-ed7b065bca42}} | r | 89d4a194657c86022d92b42f20dd227c |
As mentioned earlier, we perform our training with 924 seen tags and testing with 81 unseen tags for zero-shot settings. However, in conventional tagging case, all tags are considered as seen. Therefore, we use the 81 tag set in both training and testing. Note that, in all of our experiments the same test images are used. Thus, the basic difference between conventional vs. zero-shot tagging is whether those 81 tags were used during training or not. For generalized zero-shot tagging case same testing image set is used, but instead of predicting tags from 81 tag set, our method predicts tags from seen 924 + unseen 81 = 1005 tag set. The performances of ours and compared methods on the tagging tasks are reported in Table REF and REF for the case of NUS-WIDE dataset. Our method outperforms all competitor methods by a significant margin. Notably, the following observations can be developed from the results: (1) The performance of conventional tagging is much better than zero-shot cases because unseen tags and associated images are not present during training for zero-shot tasks. One can consider that the performance of conventional case is an upper-bound for zero-shot tagging case. (2) Similar to previous work {{cite:a051d9620639716420aa23d910cf2b4c70e12249}}, the performance for the generalized zero-shot tagging task is even poorer than the zero-shot tagging task. This can be explained by the fact that the network gets biased towards the seen tags and scores low on unseen categories. This subsequently leads to a decrease in performance. (3) Similar to the observation from {{cite:5b451368ce33fde64a6384368151d3305cc9a3dc}}, Fast0Tag beats ConSE {{cite:6cbbd254459802d3eb4fe56bf0ba5802f132d4d5}} in zero-shot cases. The main reason is that the ConSE {{cite:6cbbd254459802d3eb4fe56bf0ba5802f132d4d5}} does not use semantic word vectors during its training which is crucial to find a bridge between seen and unseen tags. No results are reported with ConSE for the conventional tagging case because it is only designed for zero-shot scenarios. (4) The baseline beats other two compared methods across all tasks because of the end-to-end training considering word vectors in the learning phase. This approach is benefited by the appropriate adaptation of feature representations for the tagging task. (5) Our approach outperforms all other competitors because it utilizes localized image features based on MIL, perform end-to-end training and integrate semantic vectors of seen tags within the network. We also illustrate some qualitative comparisons in Fig. REF .
| r | 51661e59319a5281222b9c2297fb663a |
First, general features extracted by 2D CNN in regular pixel grids with fixed receptive fields often have difficulties in handling thin structures or textureless surfaces, which limits the robustness and completeness of 3D reconstruction.
Recent MVSNet-based attempts {{cite:513096d334333af58347573614bdce9ffc5c6a15}}, {{cite:3bebb764aa1c555bd4133a8b6caef50d50e773e4}}, {{cite:5fefd2a5d614c1dac8b4b400621f911dec70b140}} introduce multi-scale information to improve depth estimation.
However, context-aware features have not been leveraged well enough for varying richness of texture on different regions.
| i | f3f4ec8853d96f5d6a62446f9786a742 |
Results and comparison: The results in Fig. REF and Fig. REF reveal that our method creates better contrast with lower noise than CycleGAN, and that contextual information from region patches assists the formation of local information (e.g. the lizard head appears much more clearly in our result). We also provide comparisons with other automated methods: DeepPrior {{cite:608269dce3aad54adb4f430091e5025e89fcafa1}} is an unsupervised denoising technique (needing subsequent histogram matching); Learn-to-See-in-the-Dark is a supervised low-light image enhancement method {{cite:d3e3ef28307ab3b5756f5e45d61aff16a364d402}}. We retrained their model with our `Static' sequence combined with their datasets. The results in Fig. REF clearly show that residual noise remains a problem for these methods.
| d | 3752f917c9202a333463b30f5f099613 |
In constructing the linear model of epochwise double descent, we assumed the existence of small scale (small eigenvalue) features that are largely unaffected by the presence of noise, and learn slowly compared to to larger scale (large eigenvalue) features which are noisy. Near critical parameterization, this gives rise to an epochwise double descent for noise levels above a threshold. This is similar to other linear models (recently {{cite:9122652eba30315e0f7b6fdb8310d47abd46b0fb}}, {{cite:f50c799dc24d7f650a67194eab0a6759da0524bc}}) that achieve the well-studed double descent in model complexity by assuming the existence of small scale (small eigenvalue) which couple to uniform noise. While not a focus of this work, we note that both effects can coexist if the small scale features are weakly affected by noise in comparison to the large scale features. We believe that the study of other noise models is likely to yield other behaviors which may be interesting, and the problem of determining which noise model is most appropriate for the case of training deep neural nets with label noise is a promising direction for future work.
| d | 02dd1d93054e85ed07a145e5c27d7107 |
This work uses the global gaussian interpolation, Gaussian Process Regression (Kriging) {{cite:d0772b829d2c453c411b7ae209b257f22d1a2d20}}, support vector regression {{cite:af86f5ecbd4d0d1934bf71ee4b366a1668a998b6}}, polynominal regression {{cite:7f2b54f582c2904ce188ce6f397fb21cf983350b}}, neural networks (NN) {{cite:62a0b322bcfdbbfda6d4ec1416dda2adc0478454}} as baselines.
| m | e9aaf6eb3e579418375901f27ad8dc05 |
Mesh-Based Approaches. Given a monocular or multi-view video, earlier methods reconstruct the detailed geometry and textures using parametric mesh fitting {{cite:2ced7d349f02aae7bb9dd74cd08c7abd34aa5018}}, {{cite:7a6f5786f8a90c0ae7f2ea23ccc7de89058d6864}}, {{cite:9cb2e48ed2b137df52b21bf12d02092f1cd8ea61}}, {{cite:3f14967b383d5823a6c97c6aefb5b98909fe4c14}}. Alldieck {{cite:2ced7d349f02aae7bb9dd74cd08c7abd34aa5018}} fit the SMPL body model to all frames and optimize for per-vertex offsets using person silhouette information to capture the clothing and hair details. A detailed texture map is then generated by back-projecting the image colors to all visible vertices. The novel views and poses of the person can be easily generated by articulating the SMPL mesh and rendering it to the image with the obtained texture map. The methods {{cite:7a6f5786f8a90c0ae7f2ea23ccc7de89058d6864}}, {{cite:9cb2e48ed2b137df52b21bf12d02092f1cd8ea61}} adopt a similar strategy but replace the optimization-based framework used in {{cite:2ced7d349f02aae7bb9dd74cd08c7abd34aa5018}} with learning-based components for faster processing. In all cases, they produce
a parametric mesh with a texture map which can be articulated and rendered in a new view. However, the main limitation of these methods is that they rely on a fixed mesh topology which cannot accurately capture complex clothing and hair geometries. Also, the articulated meshes do not account for pose-conditioned geometric deformations and changes in texture. Hence, the rendered results lack realism and high-frequency details. In contrast, our volumetric representation directly learns these details that are not captured by the parametric models.
| m | db04b43814d60ffb5c5ed66a94433ed3 |
From another perspective, we take OfficeHome as an example to visualize the effect of the adversarial training in Fig. REF . Specifically, we train vanilla ERM models (without prediction disagreement-based adversarial training) and ERM-based AdaODM models (with prediction disagreement-based adversarial training). Then we visualize the learned features using t-SNE {{cite:df9b3984bc0a1e5a3d4e01d879adad36f9a609f3}}. The left pair shows an example that trains the source model on “art”, “product”, and “real” domains, and generalizes to “clipart” domain, while the right pair tests on “art” domain and trains on the other domains. Within each pair, the two figures illustrate the source features that are generated without and with adversarial training, respectively. It could be easily observed that the adversarial training procedure successfully clusters features from various domains. To evaluate whether the test-time online disagreement minimization benefits the target feature alignment, we apply a CORAL-like metric as an indicator to measure the distance between the learned source and target feature distributions. The metric is calculated as the second-order difference of mean and covariance values between source and target features. For the above two trials, the test-time optimization process of AdaODM improves the test accuracy from 54.1% to 59.4%, and from 61.7% to 64.0%, respectively. And the metric decreases from 9.07 to 5.30, and from 1.43 to 1.39, respectively. This implies our test-time online disagreement minimization indeed mitigates the distribution gap between source and target domains.
| m | 7c36e18ecc9a509a150b8d9164c24e69 |
Hom-algebras and Hom-coalgebras were introduced by Makhlouf and Silvestrov in
{{cite:72246016fd127845e057e261b05931cf2b6d60b4}} as generalizations of ordinary algebras and coalgebras in the following sense: the
associativity of the multiplication is replaced by the Hom-associativity and similar for
Hom-coassociativity. They also defined the structures of Hom-bialgebras and Hom-Hopf
algebras, and described some of their properties extending properties of ordinary bialgebras
and Hopf algebras in {{cite:b429fffbccaf00883b405d76a5f3f1a02485536e}} and {{cite:0c5323a4242c9b522fae6ac3804f096772a23390}}.
Recently, many more properties and structures of Hom-Hopf algebras have been developed,
see {{cite:874c2efeef6c3f0c69dcc5cf2984fd9bd01b9910}}, {{cite:5a839518e29fd85f6790a57765995e8eb53c32b1}}, {{cite:b70e00203668d120f954a27cca8245dac7dfcfcb}}, {{cite:2cb2a9f1e37f8932cc9a2fe49d8a09ff8ad4cca8}} and references cited therein.
| i | 80402bc66c0a39e5568208782a2612f1 |
the first nonlinear evolution PDE solved in 1967 by Gardner, Greene, Kruskal,
and Miura {{cite:190d2f3f7f92b9275950d4a6c5cc15e10207ae97}} by the inverse scattering transform (IST). For
the reader's convenience and to fix our notation we review the necessary
material following {{cite:b1b9a3130a8f1a94012efc0a56c6b24ee7c5754d}}, {{cite:aacc7fb7d7526d9c47bf080cd4f095bfc0010a4a}}, {{cite:21fe5e38a5bf154e93be025af0d419dfa7e29958}}. The IST method consists, as
the standard Fourier transform method, of three steps:
| m | 859190d524dd7fe0becf374406319368 |
In RL, the agent is not being told new knowledge about the environment and what the correct behavior should be. The agent learns new knowledge indirectly by guidance of the environment. Moreover, this guidance is simplified into reward rather than knowledge itself. But as long as learning the limited new knowledge directly, human can speed up the learning process. This is especially important in natural language. New knowledge means new vocabulary, new expressions or new interpretations of words (ambiguity in language). The knowledge based on new vocabulary can not be learned indirectly, because words can not be created by memory. To address this issue, we merge MLE in RL to ensure that the agent can get the new knowledge directly. It is inspired by the work of unlikelihood training that the tokens updated by a likelihood can be assigned high probability{{cite:61d38c537184873e6a92040bc85e5c53468fd315}}. But our work is different from unlikelihood training. The goal of unlikelihood training is to improve neural text degeneration{{cite:8ae4220f8e132f38c959c7c4cf8dd77155faaff8}}. In practice, it regulates likelihood loss by unlikelihood loss to decrease the probability of previously generated tokens{{cite:61d38c537184873e6a92040bc85e5c53468fd315}}, {{cite:8ae4220f8e132f38c959c7c4cf8dd77155faaff8}}. And our goal is to increase the probability of tokens which are more likely used by the target domain.
| m | fd3fc87a794c4ee954a601075a083cf1 |
where {{formula:0623123e-2ed5-42a8-bcb7-3701593ce1a8}} is the cross section for {{formula:370fd1d1-09c4-4e14-8f72-b7e502f08ef1}} annihilation, {{formula:a753f3c0-98dc-48b3-b27f-82a11f824c78}} is the flux of soft photons in the observer's frame and {{formula:46fa426a-a04d-4315-a263-b1ebef5c21a6}} is the luminosity distance. The calculated minimum flux {{formula:6ac64322-fc09-4361-947e-2f73059276a3}} is shown in Table REF , and it can be seen that {{formula:c0a9be51-67ae-4367-9442-955568815d91}} for each FSRQ is lower than the flux of observational data points in the X-ray band, which is consistent with the fact that the X-ray emission from the hot corona is not directly observed. Therefore, if the jet's dissipation can occurs in the hot corona region, no {{formula:3bfee9c1-e5fd-4042-9225-2b7476f695d0}} -ray can be observed as well. On the other hand, {{cite:3f2c9a513549c993a78c109e1122e2dc14ef4c30}} indicates that high-energy neutrino flares can be produced in such a region where the jet's dissipation occurs in the hot corona, although the occurrence probability is quite low. At present, there are still a large number of jet-dominated AGNs have not been detected in the {{formula:bb7dfcca-3b51-407c-a437-d76b2834a4fa}} -ray band. If a considerable part of their jet dissipation occurs in the X-ray corona, then the jet-dominated {{formula:f4bbf9ce-551b-45e6-b7f2-ce12aada4bce}} -ray quiet AGNs may also contribute to the high-energy neutrino background.
{{table:55475b20-8715-4900-ba82-4d38d91a3f81}} | d | e2bc896749cd8edc891d4a60edd7f385 |
Given a cosmological model, eqs. (8) and (9) can be used to predict X-ray fluxes of quasars at known redshift. We compare these predicted fluxes with observations by using the likelihood function {{cite:0edeb3e0a02ea6491cdbe34f21843bb3e75669a1}}
{{formula:4b1575a6-6b90-4a42-b94b-8d467af046c7}}
| m | 42ac5a8e80c12f6076c06d7ad60cf910 |
The 3th order formalism of the WKB approximation presented in the paper {{cite:ddf77040acbed24a2292b3a2ca85bb31f8cbcbb0}} has formula
{{formula:132a752e-3261-4a31-9e1f-1d5fcaa6e0f1}}
| m | 607635a773aae21f0839ce02d45e1fd4 |
In the search for the CP-violation in the leptonic sector, crucial information has been obtained from reactor and accelerator experiments {{cite:9e96d64eb154f343ade17269198e9839a01638d0}}, {{cite:9c6c65c9476f2ad782f00ba9574fcc3f3e14e64d}}. The discovery and measurement of the third neutrino mixing angle, {{formula:93320ab8-9fee-4f25-b0cb-59d034caad76}} , with a value {{formula:adb807d2-3baa-49bf-9214-f744d3023128}} 9, corresponding to {{formula:a9702a39-bcbf-4487-bf96-34b5ea7134b9}} 2{{formula:519bc1ff-5ebe-4445-b7d4-4e52cb575c65}} {{formula:2cd4ed37-f7bd-4526-b256-157f611b15cc}} 0.095, confirmed the possibility of discovering and measuring a non-zero value of the Dirac leptonic CP violating angle, {{formula:40c9f4b9-9e78-4e08-b105-b3bed0632a85}} . Before this measurement, a significantly smaller value of {{formula:7a2eda19-03c0-4dd4-901d-b9e05ac538e7}} was assumed, with a range of values of {{formula:24d2fd77-14b6-4d52-a85a-6ae90d2670c1}} 2{{formula:450e8aab-f1f6-48fb-ad6b-7dbfe3222bc8}} {{formula:485ed0dc-9939-4583-ae22-daaf3855667d}} 0.01 and 0.09, with 0.04 as a standard value. In the light of this new finding, the sensitivity to CP violation observation and measurement, with precision, of {{formula:8ac3d3bc-ec3d-475c-b462-b8e4196fba99}} has shown a strong enhancement at the second oscillation maximum compared to that at the first oscillation maximum {{cite:79af0cf938c4d9340427869ddf3363125280a5c0}}, {{cite:dff82cae2dc9ae3e40cc368aad852e6d55903f70}}, {{cite:95522cf50775b464bec11696767cf2db1e8f7bcb}}. This can be seen in Figure REF where the change in the probability, upon changing the values of {{formula:7311289d-e4f3-4cf4-afe8-f97ec27127a4}} , is much more significant at the second peak maximum compared to the first. Moreover, by placing the far detector at the second oscillation maximum, the experiment is significantly less affected by, and hence more robust against, systematic uncertainties. This is particularly important since the improvement of the present systematic errors is known to be very hard. However, placing the far detector at the 2nd oscillation maximum implies the need to use very high intensity "super" neutrino beams to compensate for the longer baseline, hence lower statistics.
{{figure:ede947db-f5d3-43ae-98a3-75b9c8182fc8}} | i | f4d5933543a9dbd37f04f69298b469f6 |
In Ref {{cite:5658cde4b1f9789529ad75d048d8633cec3e742a}}, the S-wave tetraquark states with all quark configurations are systematically studied in a non-relativistic quark model. The parameters are fitted by reproducing all S-wave and P-wave ground state mesons. The charmonium-like tetraquark states {{formula:0d09d317-260d-4905-a00a-1a63b267d87b}} are systematically studied in a color flux-tube model with a multi-body confinement potential {{cite:79276c77fe867235006e57643b3abbfa1960cb01}}. The parameters are determined by fitting the masses of the ground states of mesons. The mass spectrum of the ground state mesons is obtained by solving the two-body {{formula:89ca80c0-b89e-4e24-a596-f7155f2d2b24}} equation. The fully-heavy tetraquark states {{formula:e1f03e21-10f7-49f1-a0c0-ef2972f30256}} are studied in several kinds of typical non-relativistic quark models in {{cite:b5bddc433aac525a77ad2abd6928343323693d45}}, {{cite:53505bcd4595b1ecc4a89d33a214b562e4234283}}, {{cite:f3a93fd120eccd2c9f37388a734eb0fd9b358e92}}, {{cite:42d7f4813286e8c13ae1b83d3bee4bde53e4b800}}, where the Cornell-like potential are considered and model parameters are fixed by comparing the theoretical results of meson states with their experimental data.
| r | 0acbe3e7446a80320d11c67a971ea1e9 |
Let us first look at some examples. Since in a complete graph every vertex subset is a general position set, A wins the {{formula:c31b39a1-2843-41cf-937b-28976d2aeef7}} -avoidance game on the complete graph {{formula:88f1e350-5bc8-4f95-a0bb-ad0d9d0b450f}} (and loses the {{formula:97bef430-c7c5-4839-8fd0-bf8f80ae2eb2}} -achievement game) if and only if {{formula:079ff5ff-47ef-4457-aaab-3f50b1af8e7b}} is even. Clearly A wins the {{formula:eb55696a-ef0a-4645-8225-2d76182c7c81}} -avoidance game on a graph {{formula:2f531c51-4614-4bf3-8999-982da2463b20}} with {{formula:7fc8c447-6f0b-4604-a333-ec809e4835f9}} . As proved in {{cite:dc9ae8392e39ec91b996e5ee7e51e01c92cdfbea}}, the only graphs with {{formula:a50212b1-a5f5-40d3-9a1f-0e7a601e8bba}} are paths and the cycle {{formula:3a19a5e6-b53b-41d6-9776-381deb4bc426}} . If {{formula:363dbff8-2b78-44a0-a4e4-98efd75c545d}} , then {{formula:e64fa108-a7b9-4130-80ec-f55640e206ec}} -avoidance game will take either two or three moves. In fact, if {{formula:ea30f868-ff85-4423-869e-044cfdebd390}} then A wins the {{formula:130bddd3-cd10-403b-abb9-d7731929402a}} -avoidance game if only only if every vertex of {{formula:ce40c904-fa95-4c23-826c-086a90fc8977}} lies in a maximal general position set of order 3. Applying this observation to cycles we infer that B wins the {{formula:930bce0d-75d0-4bbd-af96-e73ea8be4595}} -avoidance game on the cycle {{formula:6949e988-c3de-4da2-a0bd-b33f3b107d7c}} if and only if {{formula:ae43be5e-f0a7-4566-8fc6-83a37c35157e}} . On the other hand, it was observed in {{cite:7032886dfdad92b1a11789587490d5bb202ea934}} that B wins the {{formula:1b0365cb-b323-47f8-aca3-681a44c2a1f2}} -achievement game on {{formula:9a5d6888-4dc4-43e1-a10b-1c1f6a60365f}} if and only if {{formula:675b8724-a6d5-4ef6-b817-36ada3607096}} is even. Hence both the games are intrinsically different.
| r | 487dd0eb148235c0052b5289bb754aa7 |
Sketches Generated by VASkeGAN:
We allow the trained model to generate sketches after being conditioned by a sketch from a particular class. A sample of the generated images are shown in Figure REF . This confirms that the model is indeed generating meaningful sketches and not random strokes. Here too, it is very clear from Figure REF that, the `scribble effect' of {{cite:f7771232c7344a84d3303070ac563d93df5dc7ed}} is alleviated by VASkeGAN. Following this, VASkeGAN is trained as a standalone GAN wherein the the encoder and the {{formula:934a4def-5f82-48d6-b9bb-e033cee878d8}} are removed, retaining only the generator (decoder) and the discriminator with {{formula:734721f2-a2a4-418b-90e9-edaadd1ba57e}} and adversarial loss. The sketches generated in this case for all the categories are just doodles without any discernible entity. This experimentally validates the fact that discrete outputs pose a difficulty for the gradient updates to be passed from discriminator to generator for the weight update. Hence empirically strengthening the argument in favour of the formulation of SkeGAN.
{{figure:5033e860-e960-4ab1-aedb-59547ab129ec}} | r | b1955e7e76b61107332ccd25545a32ab |
Next, we give an overview of our approach.
For any given query point {{formula:0d03ca03-a8ac-4f11-ba76-7244fdb6eb68}} , we compute two binary trees (called visibility trees) to store the visibility polygon of {{formula:abb4182e-b815-4c60-818f-85fe608327df}} .
These visibility tree data structures in our algorithm are a modification to visibility trees defined in {{cite:9e8743ee82ab7a5d78374b400beb1464b7f320cc}}.
From these trees, we compute the visibility polygon {{formula:a183c7ab-9c3d-4495-bcbb-3aebe02f0b91}} of {{formula:f74bf036-a38b-456a-9792-dddaec640afd}} in the current polygonal domain {{formula:52f9ae33-cf21-4a8f-a70e-7b65a1187497}} .
Whenever a new vertex {{formula:9c697552-d634-478c-8602-8cc706ce8186}} is inserted to an obstacle of the current polygonal domain {{formula:64382a78-e73d-4001-aa18-dfced6af3b20}} , first, we perform a ray-shooting query to determine whether {{formula:1ecdbfed-7383-42c0-bc31-445283ce787e}} is visible to {{formula:7f8e5617-fcbe-4cb7-af41-6a12d16121fa}} .
If it is not visible, then our algorithm does nothing further.
Otherwise, we traverse the visibility trees in depth-first order to determine all the vertices that are not visible to {{formula:be4daa65-c608-43a0-9fd9-941cae8bb303}} due to the insertion of {{formula:ec9a7fb8-8370-4f2e-ad0e-cb05b7ab3d30}} .
We update the {{formula:e49e7da1-1e4f-4916-add3-9e85afffdd1f}} by removing these vertices from {{formula:bb5b7a79-836a-4b7b-9f3b-56041186d508}} .
As part of this, we also update both the visibility trees.
Similarly, in the case of deletion of a vertex {{formula:629ae192-a57d-491a-aa9b-fc583dd26399}} , by performing a ray-shooting query, it is determined whether deleted vertex {{formula:73c021da-7e97-45d5-a8c2-3ca905159ec1}} is visible to {{formula:f0f07848-ec5b-4984-be37-845c4916f970}} .
If the deleted vertex {{formula:23fa0d41-3257-4ef0-8160-d54ccebefa67}} was visible to query point {{formula:9ce47139-8f47-420f-8b93-b6bc253f13fa}} , then the deletion of {{formula:3c9c181e-cd15-4111-8af0-9d0416544874}} may cause the addition of some new vertices to {{formula:94b8e479-7428-4cb7-b1b2-371015adca1c}} .
These new vertices are determined using our output-sensitive visibility polygon query algorithm with {{formula:5b63b507-12b0-4cc7-8912-824a926e0ea6}} as the query point.
Updating {{formula:dd6f22e5-48c0-4f78-b28b-81824fb0bce8}} involves merging a set of polygons due to these vertices into data structures that store {{formula:523268c9-1be9-463d-87cb-f23caa70f702}} .
Our visibility polygon query algorithm enhances data structures designed in {{cite:48eaeac5aa20096a3c72ca377502ee7289c04f10}}, so that they work efficiently among dynamic polygonal obstacles.
As in {{cite:48eaeac5aa20096a3c72ca377502ee7289c04f10}}, our visibility polygon query algorithm computes visibility trees of a query point {{formula:aeae1420-0846-4ea6-b987-97a199de0a80}} by determining sequences' of traingles bounding rays of visibility cones initiated at {{formula:60d7ce8a-9987-42ae-9809-4fa5c77333e8}} intersect.
When any ray in a visibility cone is found to strike a point on the boundary of any obstacle {{formula:2a12cc44-7495-4e14-89a8-db8aec17d8f9}} , we compute tangents from {{formula:3c15bd80-de62-4389-b855-4b250b0da8ba}} to {{formula:fa4fbb53-5b4f-47ab-abd3-a3a1a78555c0}} .
To compute these tangents and, in turn, for updating visibility cones, our algorithm dynamically maintains hull trees of obstacle boundaries using the algorithm by Overmars and van Leeuwen {{cite:5c8fc470c8762ea5e0e69f7916f3b4ecb57f3038}}.
In updating the visibility graph, we use an important characterization to determine edges of the visibility graph {{formula:55b27a3c-f620-467a-ba25-7e5bb5c53185}} of the polygonal domain {{formula:d03dbaca-a309-4eca-afca-433f3f71d6ba}} that need to be deleted from (resp. included into) {{formula:5606efe3-1b2d-4b97-a14e-3dee25661197}} due to the insertion (resp. deletion) of a vertex from {{formula:8587671c-9c97-449b-8e71-b61904cc52c8}} .
| r | 900c98f8b7c1ef0d922c52c520ab6f9d |
Auxiliary variable approach. The proposed schemes are based on a reformulation of the time fractional Allen-Cahn equation
by introducing an auxiliary variable — an approach intensively studied recently for gradient
flows; see, e.g., {{cite:a665ced9d4dd6a4910704e0dbf5060e5dc1a27e2}}, {{cite:a3261249344703d7074860ea8c6e52a80222bf8f}} and the references therein.
The key to is to rewrite the original equation (REF )-(REF )
into the following equivalent form:
{{formula:43085b7c-fa31-4357-b3dc-cc2f4bdc2e60}}
| m | c75c9e402f23a9d5f23b7d33b172f77a |
where {{formula:e57589ff-b27f-4261-b0ad-a120f770cb71}} is a flat measure, the standard deviation {{formula:9c1bec36-fb2f-44d6-89bb-ddf0a333db14}} sets the energy units (and was set to 1 in the numerics), and {{formula:92e74d3c-4647-4a0f-8c46-22dab6dc4f4b}} is a complex hermitian or a real symmetric matrix depending on whether we work with the Gaussian Unitary Ensemble (GUE) or with the Gaussian Orthogonal Ensemble (GOE), respectivelyThe third canonical Gaussian ensemble, which we don't study in this article, is the Gaussian Symplectic Ensemble (GSE). It addresses time-reversal-invariant fermionic systems displaying Kramer's degeneracy. {{cite:518edbfbb347aa0dcf302e337055731706b99422}}.
| r | 0ca9b8a0f29309598443e80340b9d27d |
We have several observations from the Monte Carlo numerical experiment.
As shown in Figure REF , all methods are outperformed by the proposed identification scheme, which maximally incorporates the internal positivity side-information.
Indeed, the side-information helps excluding spurious model candidates and subsequently increases the accuracy of the estimation. The bias-variance results presented in Table REF confirm this fact.
While each of these methods estimates a non-negative impulse response and partially integrates positivity side-information, we can see that the level of integrated side-information is
less than that of the proposed method.
The proposed approach incorporates this information in the model
maximally.
Comparing methods B and C, one can see the former one is a two-step procedure where the estimation is performed in the first step and non-negativity of the impulse response is obtained in the second step, while the latter approach is a single-step procedure that considers impulse response non-negativity during the estimation.
On the other hand, according to the fitting results shown in Figure REF , C performs better than method B. For methods D and E, we have similar arguments.
This observation highlights the importance of jointly considering the positivity with the impulse response estimation, as done by the proposed method.
Method A knows the actual order of the system. However, according to the results presented in Figure REF and Table REF , one can see that positivity is a more advantageous and stronger side-information for impulse response estimation, especially when it is incorporated with its maximum strength like in the proposed scheme.
Finally, one can see that the kernel-based methods D, E, F, and G have better estimation performance comparing to the methods A, B, and C, which is expected {{cite:ffb104a84a39aae662755478af371b2ec6cc1896}}.
{{figure:c5fc3542-0186-4fa2-a7be-c929e4a7c8b9}} | d | e13ba92c1f985a6adaca73feb216847d |
The unpolarized and polarized non–singlet and singlet anomalous dimensions have been calculated at one–
{{cite:e89665dc21df55ca3d359956d08b9fc772426ee3}}, two– {{cite:25d69d42f32fc71b1975768945c9ad55f5ab5855}}, {{cite:d3a59ab683ea2dd2a0e58e2dfd64e4f9ac802f3d}}, {{cite:e02ffc019935c5fc05c23c2e8962c9ff587e0ac8}}, and three–loop order
{{cite:d82a2020861d6dd3ab54334eef870bf996fc6eb4}}, {{cite:245badfb5a2afb8ac452b7d8c95610f858d5c217}}, {{cite:6727eb4e1d31a20b4a4a06be8e37b4130fc0221c}}, {{cite:9ca42dac4a1d0810e0c5f6cd979adc17341f696c}}, {{cite:ac1b1e4ff6a79cb6e70384755c106ded34e59dfb}}, {{cite:9bf226b74c2831aa4e308306991713f34acf027f}}, {{cite:f8ac235b73d878596c929ad639ac8563dc2c2c3c}}, {{cite:b33a0a55cdc5c932f44d9f96eb28de957684f549}}, {{cite:900617b0db5ac729c02ddc953e9cbef7130863ab}}. Here different techniques
as off–shell massless OMEs, the forward Compton amplitude, in part also with scalar and gravitational currents,
and massive on–shell OMEs have been used. In the latter case one obtains the contributions {{formula:ec211076-a0c6-4d24-a202-1deee02562b4}} , which
are the complete anomalous dimensions in the cases {{formula:7bca9228-54f4-4aeb-9f77-5190c6015845}} and {{formula:a7a87d8b-ff32-41f9-a826-00a99f59e4a0}} .
| i | 62362e7cfe37d5fe111aaac26a55c99b |
tocsectionReferences
Appendix
Robust Central Path
The goal of this section is to analyze robust central path. We provide an outline in Section REF . In Section REF , we bound the changes in {{formula:fbf2ee60-f696-4ca1-869d-f9f71ca2e2b2}} and {{formula:fa935e72-1fbd-4788-b018-6532b0c02ebe}} . In Section REF , we analyze the changes from {{formula:b7d2a491-b810-4eea-866d-92fbd5b9a45b}} to {{formula:d515147e-835c-4ec9-a836-de6d037d5c4e}} . In Section REF , we analyze the changes from {{formula:0c6a29e9-5a4a-4ba9-88eb-a74b16014054}} to {{formula:f7444c30-7229-49c3-b9ad-833cb784c101}} . We bound the changes in {{formula:a7a09001-7fac-4b5e-8c8f-456cbb4688cc}} in Section REF . Finally, we analyze entire changes of potential function in Section REF .
Outline of Analysis
{{table:e4695c5b-7b97-4a58-84f8-2bd42440e417}}Basically, the main proof is just a simple calculation on how {{formula:44666881-f122-4b72-9cd2-146e565b4efe}}
changes during 1 iteration. It could be compared to the proof of {{formula:5d102780-ea6f-46a4-80c0-2d636cfc9f54}} potential reduction arguments for the convergence of long-step interior point methods, although the main difficulty arises from the perturbations from stepping using {{formula:53358e6a-1099-4eee-a736-0d654daad65b}} instead of {{formula:781dd2e8-b76f-41e8-9de9-20d067a2813e}} .
To organize the calculations, we note that the term {{formula:7d39fc13-f65c-4a3f-85b8-9594c0b7028e}}
has two terms involving {{formula:41975255-1262-49a3-9a90-22e71d8dcf6c}} , one in the {{formula:5814f701-fffd-4036-8abb-3f66288cd455}} term and one in the Hessian. Hence, we separate how different {{formula:80f36b20-f78f-4df9-98b1-9676cd2279d6}}
affect the potential by defining
{{formula:17bc0754-ed16-46aa-85eb-fa45957e737c}}
One difference between our proof and standard {{formula:1c43318c-c2de-4ebe-b3ec-3cfef8313d4b}} proofs of interior point is that we assume the barrier function is decomposable. We define {{formula:256c002f-208a-4aaa-bc4c-feb23aec6ac5}}
is the “step” size of the coordinate {{formula:00fb7d6f-7595-4dd8-816d-cd6a78bcb1c7}} . One crucial fact we
are using is that sum of squares of the step sizes is small.
Lemma A.1
Let {{formula:3d96a90f-51b1-4e61-9ded-2282c4a36450}} denote the parameter in RobustIPM. For all {{formula:89579c9f-f76a-4f35-9080-e249232f53fe}} , let {{formula:d571514a-686d-44cd-b377-04cf5f3d6d0a}} .
Then,
{{formula:1899f415-9ae7-4c82-8723-bfaf214b30bf}}
Note that
{{formula:b44f877d-23e3-4f01-80d6-3b05d796cf45}}
Since {{formula:a8a9fa10-b360-4406-aa29-029c1d9617b5}} ,
we have that
{{formula:6f05d09b-e9e5-4b6d-a6dd-6599aba67ff9}}
Using {{formula:67066bed-19a9-4d16-9dbb-7f7b1726a19a}} , we have that
{{formula:6e71175b-0746-4cf4-ac62-c30ee1cc3e5e}}
where we used that {{formula:1a604319-f7bc-4c06-8edc-1c1e85d68eb7}} is an orthogonal projection at the end.
Finally, we note that
{{formula:c6239a43-f84b-447f-ada9-399669e5ea00}}
where the second step follows from definition of {{formula:7fe9e5ae-f228-4227-ad40-a2df2fc59245}} (REF ), the third step follows from definition {{formula:188648d9-bb07-4818-a779-0646344d7036}} (REF ), the fourth step follows from definition of {{formula:51cd1168-f057-4941-90dc-4c0381ca4c27}} ().
Therefore, putting it all together, we can show
{{formula:9e6238c9-e39b-44f4-a6ed-bb858499e7b2}}
Changes in {{formula:35bfc586-fed7-44a7-a9ab-0afcf6670442}} and {{formula:7286cd08-b4a5-462a-af10-b6e5e3cfb5ba}}
We provide basic lemmas that bound changes in {{formula:bddba249-b928-49aa-81c7-e671dacb1a34}} due to the centering steps.
Lemma A.2 (Changes in {{formula:886a8e29-af92-4c59-a928-ab56c357416d}} ) For all
{{formula:8f340a2f-e8b9-4b9e-961a-13f0424f6355}} , let
{{formula:dd581455-7513-49fd-86ae-8f59e3258878}}
Then, {{formula:5a30ea97-eebd-461b-b96b-0eb785cfe571}} .
Let {{formula:bf7c881d-7c69-43ca-a7bc-64e0d0aaa740}} and {{formula:cd6826d0-5651-4160-91d7-680a610c05e4}} .
The definition of {{formula:b9f92eff-6d15-4d54-b244-14d76a9141a0}} (REF ) shows that
{{formula:2e030dda-6f3d-4df2-8a5a-50b428dae5bf}}
By the definition of {{formula:9fa6f7d9-c3b6-4a7d-9cfb-c6cb795c9632}} and {{formula:1eca0af5-d4f8-4b57-b26c-b4f6b912ba77}} (REF )
and (), we have that {{formula:b4233904-bf70-4c0d-aa5b-87a0b0aec793}}
Hence, we have
{{formula:f8e433d6-7136-437f-8378-456d406e9eaf}}
where
{{formula:d6fe11d4-b2f6-4c33-8e58-f775ecd98431}}
To bound {{formula:d01c7d34-2af0-42bf-bf95-fb367fe8c0b7}} , we note that
{{formula:c000af2a-3874-49ae-b2f4-0c7d58cec4f1}}
where the first step follows from triangle inequality, the third step follows from definition of {{formula:3d021b37-d535-4b0a-865d-e0fb23d2dcaf}} (Lemma REF ), and the last step follows from {{formula:48383b3e-c6e7-48f9-881c-329533c79987}} (Lemma REF ).
Using {{formula:dc220fbc-e30b-4417-a56c-c259877d0532}} ,
Theorem REF shows that
{{formula:78381f62-f80e-4506-9ed1-4ec398fddb5f}}
Equivalently, we have
{{formula:08997f22-0339-48af-9734-518ba76075e0}}
Using this, we have
{{formula:6d63f449-dd56-44f1-a6c8-d1022e056cb2}}
where the last step follows from definition of {{formula:90f7d7f6-3ce3-4e8b-9ed6-386dd3f6572e}} (Lemma REF ).
For the other term in {{formula:76837df9-7079-41bf-aed3-741d843c439f}} , we note that
{{formula:8b4a1bf4-6ff3-4973-9c60-f91f93d9be8d}}
Hence, we have
{{formula:b67c8351-206f-4b9a-a370-6cbbb4f53ad3}}
Combining (REF ), (REF ) and (REF ),
we have
{{formula:ee4cdca8-b50e-460a-b21a-bb9cbf3155ca}}
Finally, we use the fact that {{formula:da62c482-fbca-4a74-86e5-596b2b52657e}} and {{formula:baf425e5-dc75-4404-9973-50160f436c6b}} are
{{formula:38799454-f93d-487b-985c-b38f8a4595ad}} close and hence again by self-concordance, {{formula:28be017a-d9ab-4653-8e9e-ae63bed3247a}} .
Before bounding the change of {{formula:b95233c0-df19-4a7b-9668-8929c0232eca}} , we first prove a helper lemma:
Lemma A.3 For all {{formula:5aa86ff0-ef17-43b7-a0e7-bdd8b287c777}} ,
we have
{{formula:32698cde-ed73-411f-85c6-a07bc67b9875}}
Note that
{{formula:aca54650-d7d3-4de3-b357-8566a020fbe0}}
For the first term, we have {{formula:22a3f2a5-c428-4e3a-9d04-3cfa659b9d38}} .
For the second term, let {{formula:b233b669-1eca-43af-bb11-2711b4e3f6f4}} .
Since {{formula:6e4188e3-df7b-4f73-867d-9e463383d300}} is close enough to {{formula:9b1d2c5f-26e9-411c-8e46-84d9fd546e7e}} , Theorem REF
shows that {{formula:5adf2c95-ae2e-40b1-ae24-fc551e6546d7}} .
Hence, we have
{{formula:00c21597-1011-4779-8101-78bbbba8be76}}
Hence, we have {{formula:23b85076-20bd-4635-bf28-7a73bcc97053}}
and using again {{formula:751611ce-ef14-49ca-b980-abcbba1f3d41}} is close enough to {{formula:73c7f165-45ab-4aa7-917b-e4429cad9cd4}} to
get the final result.
Lemma A.4 (Changes in {{formula:b5894c87-a7a1-49c3-8586-61e8e7fb9785}} ) For
all {{formula:165dd44d-2842-4c7f-8cd4-af4aa603bc98}} , let
{{formula:d5481841-2100-45f1-90f1-5f567bb872e1}}
then {{formula:43530a7d-5805-499e-82e7-86aa50bd22f6}} .
Furthermore, we have {{formula:0bc8cf13-3072-4153-8b3e-817c5399b471}}
For the first claim, Lemma REF , the definition
of {{formula:b3c4ee5d-917f-4b15-b468-355a3a9c6512}} (), {{formula:d9ad6ca9-8818-4704-98bb-de16f084fab6}} (REF )
and {{formula:41e21750-c983-4d43-bc1f-e413151cfc48}} (REF ) shows that
{{formula:16ca71f0-69ac-4179-9014-b31b8964aab6}}
where {{formula:07f1a755-f5ae-448c-92d0-42a2cfa28aea}}
From the definition of {{formula:4b69aac8-4b8c-44f4-8277-a778c57b119b}} , we have that {{formula:6852df0c-c392-47d4-939a-54cc063fe16f}}
and hence {{formula:89c1d49c-80bc-4929-867d-cb904bb5c8e7}} .
Therefore, we have
{{formula:71feeb1c-7f12-493f-b8eb-6697fba9ee32}}
Now, we bound {{formula:9d11e2d4-018a-42b8-9abf-31cd0eba3a28}} :
{{formula:d4bef475-e56a-4b9d-9f80-cc9dfd253df4}}
where we used Lemma REF and Lemma REF
at the end.
For the second claim, we have
{{formula:d4cc9f65-7f2c-461a-a7af-f0ed11f110f3}}
where we used (REF ) and that {{formula:e31fe320-23f6-4362-950b-06fa9bd8ba0c}} .
From Lemma REF and that {{formula:81741725-373b-42db-aa26-516019f34ed9}} ,
we have {{formula:8191ded2-faaa-4715-a2d5-a5e4f82f3bf3}} .
Movement from {{formula:01f083bf-8770-4ab4-a2db-0a24c0b0769d}} to {{formula:12a2f65f-65c4-42fc-a1d7-2e9e643b3803}}
In the previous section, we see that {{formula:bff205e8-15fb-4481-a352-3242a043a6af}} will be expected to decrease by a factor of {{formula:17a2cdad-7ce2-4898-b775-ba944b339b85}} up to some small perturbations. We show that our potential {{formula:9bc627fe-2195-49db-8765-8783c6be09f4}} will therefore decrease significantly.
Lemma A.5 (Movement along the first and third parameters) Assume
that {{formula:a98cb441-f59a-47e6-8f79-425bcff98af2}} for all {{formula:e9dcf995-1826-498c-b76b-46a722b3c626}} . We
have
{{formula:414734af-2a1c-479d-ab51-95b55d1ea476}}
Note that {{formula:fadbcd22-61ca-4d49-9df8-32c01b3aafca}} is a function that has three inputs. We use {{formula:be59eba3-be54-42d4-afba-1bf4f9ddb3d6}} to denote {{formula:cd83bdab-8757-466a-94c1-74b47230811c}} for simplicity.
Let {{formula:0643bcfc-569c-4627-bfba-71ec11a36b3f}} , {{formula:4fbe2868-21b5-4713-8771-206a54065980}} ,
{{formula:46fe09d7-a019-495e-acd6-1ebdd26dee70}}
Then, we have that
{{formula:18e425b8-76ea-4ae8-8ac8-4e086c12bf7d}}
for some {{formula:7d799d28-30c1-4ed7-989e-b41e720d2199}} . Let {{formula:c96b31da-c581-4439-aec3-2a918adbec67}} .
Lemma REF shows that
{{formula:1c638d86-c3ce-4e33-a3ed-db667ed4d164}}
and hence
{{formula:06a40736-c143-4ed7-b877-52857e360ae7}}
To bound the first term in (REF ), we first relate
{{formula:ac33a03b-9e84-4bca-b2b1-becbc8fcc2be}} , {{formula:13854ecb-b031-4d32-8cb0-25c43e771145}} and {{formula:096097c6-3979-40b0-9b57-424df2cdf594}} .
Lemma REF shows that
{{formula:c549f80e-f049-4ff0-8237-d7fd68c4f190}}
Finally, we have
{{formula:8471165e-cbee-4e03-a6ea-cd56af9c9c33}}
where the first step follows from definition, the second and third step follows from triangle inequality, the fourth step follows from {{formula:b11e7386-d006-4204-bf1b-133e1189917b}} , the fifth step follows from self-concordance, the sixth step follows from Lemma REF and that {{formula:d32ad4cf-2b01-45a2-95d8-40818032ec66}} for all {{formula:1f8ab3cc-1d99-4cd1-870c-00db8dac2e4e}}
Using (REF ) and (REF ), we have
{{formula:f03eb80a-07ac-4646-82ec-81a80eca9f4b}}
where the third step follows from {{formula:75c221b5-bcaa-4aa5-85fd-9e11574a012c}} , and the last step follows from (REF ).
For the first term in (REF ), we have
{{formula:af06a959-8664-493d-8ec5-9393ba5ec46b}}
So, if {{formula:1b6f2a32-93eb-40b0-8500-250fc2db2794}} ,
we have
{{formula:9f4238b6-0641-4adc-ab2e-2adb230559f7}}
Note that if {{formula:4eff0091-fe7c-4f5f-9421-856d71f3e67e}} ,
this is still true because left hand side is lower bounded by 0.
For the second term in (REF ), we have
{{formula:acbe28e6-c926-4f25-bca8-56f435c53c30}}
where the second step follows {{formula:2c710b61-cbea-4dc3-ad0d-e56b7a51470c}} , and the third step follows from each term in the summation is non-negative.
Combining the bounds for both first and second term in (REF ),
we have
{{formula:223f9205-e8c4-457f-9518-af7b8f2dfe23}}
where the last step follows from {{formula:6c3faba3-684f-4f9c-a7b7-29d46a3b8479}} .
For the second term in (REF ), we note that {{formula:eee4f0b4-76ef-4e63-88b2-2d31a93ddcc2}}
by (REF ) and (REF ). Hence,
{{formula:ec74189a-9d02-4b9d-8b77-0affa9e187af}}
Now, we use {{formula:6846c57a-3e3d-41b2-aac4-bbc2d4cd5920}}
(Lemma REF ) to get
{{formula:dbca0b50-a7a7-4472-a8ab-3be36f29bd0a}}
where the last step follows from Cauchy-Schwarz inequality.
Note that by using Cauchy-Schwarz,
{{formula:051e5f39-720e-46fc-afd4-580436a31f20}}
where we used the definition of {{formula:1ec7cef7-b2fc-444a-8567-7ced5d8f4ec6}} , Lemma REF
and {{formula:4ef50d10-ac7a-4ec5-a6ae-8345fbce1d70}} . Together, we conclude
{{formula:403ac54d-e058-4528-8570-6900ce980cf6}}
Combining (REF ) and (REF )
to (REF ) gives
{{formula:2bcbbf30-496c-414c-b9e9-b7c911b6c158}}
where the last step follows from merging the first term with the third term.
Movement from {{formula:c3f4c1fa-1c03-4b3b-b34b-31ef7496558a}}
to {{formula:7e531091-d5e5-46c5-97ab-0f90d3fa934c}}
Next, we must analyze the potential change when we change the second
term.
Lemma A.6 (Movement along the second parameter)
Assume that {{formula:a665ebad-56de-43ab-b99a-5c3ec4c15265}} . Then we have
{{formula:11e4a684-ead1-4cc0-986f-b39b89621de6}}
We can upper bound {{formula:f0a75f37-2ed8-47c3-82ab-7b250f2368ae}}
as follows
{{formula:f441a443-688b-4e9f-b012-9b1f7e1c8ada}}
where the second step follows from {{formula:17d9eeeb-7182-482e-abf6-fcbdef887953}}
by self-concordance (Theorem REF ) and {{formula:3572c9e7-d9ca-43ad-8675-b76a2fa457dd}} .
Now, by Lemma REF , we note that {{formula:65f425d2-840b-4d01-95f7-4b95681e6403}}
and that {{formula:84de55c8-d37a-4188-878d-f83e167fbbb6}} . Hence, by a simple taylor expansion, we have
{{formula:a6d63b2c-38a0-4c21-9e8e-fc0f20195182}}
Finally, we bound the last term by
{{formula:1a1cb774-c547-4711-9437-ef7d9cdf6a95}}
where the first step follows from {{formula:4533299a-339c-4ef7-96c5-dfb7f32862e1}} , the second step follows {{formula:e084c842-b942-4873-94db-f676a28e665a}} , the third step follows from Cauchy-Schwarz inequality, the last step follows from {{formula:40ad5df7-40dc-4b22-9866-44d64bce2acd}} .
Movement of {{formula:c845b724-096b-4b57-bd1d-c8c0aadf9829}}
Lastly, we analyze the effect of setting {{formula:ac3c6e46-3a9c-43c9-b932-2d7ce7b28e74}} .
Lemma A.7 (Movement in {{formula:3e6c194d-54f2-4902-8d77-cc464a28d684}} ) For any
{{formula:41e133a5-9e64-4e23-a031-40c95149f70d}} such that {{formula:4c9ec1ee-1e8a-4a25-b9b1-253e54a9f90a}} for all {{formula:8b78acc5-f9bd-4dfa-8909-061932e2127b}} , let {{formula:792e6d8f-716e-42fb-86ab-fdb1b72c4500}} where {{formula:ebd869ab-ff33-455a-9186-5b2b43e26d6c}} ,
we have
{{formula:412c7aa5-a3bf-4433-b42f-4936582fde2b}}
Note that
{{formula:50cc54d7-57dc-43ef-8225-8292261286df}}
where the first step follows from definition, the second step follows
from {{formula:71e27716-cc6e-41b6-b2e4-956d47d7114f}} , the second last step
follows from the fact that our barriers are {{formula:848f75f9-7d30-4bab-9815-63ae0eced37a}} -self-concordant
and the last step used {{formula:66a10b39-0c16-4a16-8d16-966e656a7026}} and {{formula:92787295-6c7e-4b55-9957-40da284c8cc0}} .
Using that {{formula:025c87c2-071f-4fa6-80a7-abc6a4bb8518}} and {{formula:638c532d-3791-4ddf-9c89-8c922cc79084}} ,
we have by simple taylor expansion,
{{formula:f90cf5df-a2c3-4295-bee3-b8abb0a661a9}}
where the third step follows from Cauchy-Schwarz, and the last step follows from {{formula:68b47048-2e80-4dde-b006-7ba89cdd2ff6}} .
Potential Maintenance
Putting it all together, we can show that our potential {{formula:667090dc-80df-4aa3-b555-d7405b458787}} can be maintained to be small throughout our algorithm.
Lemma A.8 (Potential Maintenance) If
{{formula:c27e6b36-20d9-4dda-bf97-d4202859df9a}} , then
{{formula:fc5db435-a45d-4223-843f-5aa908311be3}}
In particularly, we have {{formula:fd793105-8753-4ed6-853e-08df905bdbba}} .
Let
{{formula:96a8c71b-4195-446f-b2c4-d1b4c8315102}}
By combining our previous lemmas,
{{formula:71f56a35-412a-4a62-b017-df52035e0c77}}
where the first step follows from Lemma REF ,
the second step follows from Lemma REF , and the
last step follows from Lemma REF . We note that
in all lemma above, we used that fact that {{formula:490a2234-1fa6-4154-bb8b-31b29fc8753a}}
(for different combination of {{formula:f56a4fe0-e19a-4d6d-8cfd-105a6e0f636f}} , {{formula:abcfb9ba-546f-4ae0-89b8-5b2286932cc5}} , {{formula:b975bf77-2ebb-44ee-afef-5e3a3a562fd9}} ,
{{formula:dad9d316-05c8-444d-820d-06e74eb85434}} , {{formula:ac78c22e-b056-4e08-babe-8d43e5aeaf38}} , {{formula:9da1aef0-6624-4a33-a315-46830c62478c}} ) which we will show later.
We can upper bound {{formula:85ec00c4-92b1-4756-abd1-4446e05e6a7c}} in the following sense,
{{formula:4e9ed11c-0575-44bc-bc51-2cf911ed893e}}
where the first step follows from self-concordance and {{formula:6b95700d-78a3-4a00-b49d-ded974f2988f}} , the second step follows from Lemma REF .
Hence, since {{formula:7e4349c5-a7d1-4114-b90a-40b2bcac49b8}} changes multiplicatively when {{formula:618e7b0b-4a61-42eb-976a-f2f242cc280a}} changes additively, {{formula:ef0aaccd-1268-4bbe-9d96-b577a4468a96}}
Lemma REF shows that {{formula:d0d279ac-3db3-4ecd-b387-4e143a31fa8d}}
and hence
{{formula:93a9174d-69b5-4db6-9088-8ef2634101a5}}
Combining (REF ) and (REF )
into (REF ) gives
{{formula:86088a34-3c85-4ca9-9699-e5910cc988ea}}
where the last step follows from {{formula:9bc88be3-1d5d-4ab8-8773-00a16b21ae5e}} and {{formula:53c4734e-d3b6-4141-96c0-090c5cce065a}} .
Finally, we need to bound {{formula:cc7be8ea-4e65-4af5-9b70-c64c850c8ec4}} . The
bound for other {{formula:fb12ed44-dfae-4918-8e43-062d2b1754b7}} , i.e. for different combination of {{formula:38d3d4fe-1ce3-4e73-af4b-0f0ed20c170f}} , {{formula:4031fb5f-162f-41c2-8f7a-ece566b5eacb}} , {{formula:def30616-1cc7-44e9-bdf0-10b01560c9cb}} ,
{{formula:916afc5b-418e-4f90-bcf7-fb91be9fd782}} , {{formula:813ada24-3353-44bf-a8ba-c0dd9de5db09}} , {{formula:6b6b5ba2-2460-4b89-ab44-2f534c5c86eb}} , are similar. We note
that
{{formula:edb79aa7-4059-401f-b119-cfda146be550}}
implies that {{formula:ae96e0df-ddc3-453d-9681-f8c434e366d4}} .
Hence, by our choice of {{formula:8c738961-241a-4a5e-9f6a-0527b2e59b8d}} and {{formula:9d460a70-e8ad-44e3-a6b1-d8839b7e4cba}} , we have that {{formula:12b92695-6c92-488a-a42d-96680f89c4e9}}
and hence
{{formula:dcf90286-36e3-4841-af9c-0c62570f7346}}
Finally, using {{formula:4eb122c7-bedd-45c2-be23-5f708b3ca261}} , we have
{{formula:24b42e51-2351-4841-91d0-dd9326bae9a9}}
Since {{formula:36e537bd-f25f-40b5-84eb-3bc44a94ed45}} , we have {{formula:5fbd2c1f-b7b6-4ed3-b165-a3af8f7c5edb}}
implies {{formula:829609e3-428b-4d74-9c35-9b6b9062b0c9}} .
Central Path Maintenance
{{table:a186dffc-2718-4d88-898c-787d7f72b00a}}The goal of this section is to present a data-structure to perform our centering steps in {{formula:e33f330a-9268-4940-99e4-21b27374cf16}} amortized time and prove a theoretical guarantee of it. The original idea of inverse maintenance is from Michael B. Cohen {{cite:a1e4f138406dac63ce8e5fbd2286d807c7777637}}, then {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}} used it to get faster running time for solving Linear Programs.
Because a simple matrix vector product would require {{formula:5e4ef5b5-99b7-4e5b-9014-d7226e63842f}} time, our speedup comes via a low-rank embedding that provides {{formula:97089954-6a2b-4e26-b8c5-facb93625609}} guarantees, which is unlike the sparse vector approach of {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}. In fact, we are unsure if moving in a sparse direction {{formula:24dac53c-0bf7-49c3-8c08-89dc4274a619}} can have sufficiently controlled noise to show convergence. Here, we give a stochastic version that is faster for dense direction {{formula:dbe5d792-18d3-4cc3-9d39-43f82db85d23}} .
Theorem B.1 (Central path maintenance)
Given a full rank matrix {{formula:15722736-8c77-4b0c-9ea8-1472a696b964}} with {{formula:bdb3b4df-9521-4661-84e0-b1dcd9102a84}} , a tolerance parameter {{formula:8dbee98f-3880-4ccd-ab3d-e901312ea739}} and a block diagonal structure {{formula:9956b5fc-939c-415a-86a9-2110d5e99439}} . Given any positive number {{formula:23e9bb09-e4e9-4422-a9f9-5ccbf0734c1c}} such {{formula:7cabf15d-16b8-4617-b3e3-5996cacdaec6}} where {{formula:e790fafd-1520-4d36-a49a-47c16d31dd7c}} is the dual exponent of matrix multiplication. Given any linear sketch of size {{formula:5bfb430a-e0b4-458b-abc0-1900b0fc608a}} , there is a randomized data structure CentralPathMaintenance (in Algorithm , , ) that approximately maintains the projection matrices
{{formula:d917f248-3fd0-4152-ab15-017ae906e8da}}
for positive block diagonal psd matrix {{formula:27ca24ae-9ba8-4d68-9be4-20b72fd85954}} ;
exactly implicitly maintains central path parameters {{formula:e16c9620-6e77-49b8-a870-cc6b6b58726c}} and approximately explicitly maintains path parameters through the following five operations:
1. {{formula:473e11c2-a80b-4e9b-bfd3-e73d68f4f794}} : Assume {{formula:8bb3fcb4-01b3-4bda-bea4-05b5b2ce84c6}} . Initialize all the parameters in {{formula:252a88ad-869f-43cd-8423-bb7864ca985f}} time.
2. {{formula:b00798f1-8e2e-4c9a-a6da-b92f6ef52d87}} : Assume {{formula:00457279-0240-4ae3-9c65-eca2eed34f18}} . Output a block diagonal matrix {{formula:91a3fed7-f943-48b9-8749-cf0ad1ed232a}} such that
{{formula:652524ad-39a3-4683-96e8-76917dddb939}}
3. {{formula:f7327862-4e79-4317-a96f-851263d9a31e}} : Output {{formula:696c4217-f080-4298-85d7-5eff7cd12764}} such that {{formula:6179f3b6-c733-4cb0-bd83-f5e30cc63547}} and {{formula:52f70927-3dd7-4ad0-93c6-01228debd14e}} where {{formula:164d6de8-3045-433b-b46f-d74be2b22ac6}} is the last {{formula:f3fc3139-714b-4c42-a842-515aaf9995fd}} used in MultiplyMove, where {{formula:ac3b4b80-9f0c-4ae4-9e21-9782d265a685}} and the success probability is {{formula:80acd057-3c2b-4392-82d9-7a307c4f48e4}} . This step takes {{formula:077265f0-57f7-4069-af89-c64b45d823aa}} time.
4. {{formula:7b61264d-9f3a-48ed-8f71-2d6259ff1c8e}} : It outputs nothing. It implicitly maintains:
{{formula:393c629c-246b-4c59-bf77-25214eda24b8}}
where {{formula:58ae7664-7ec0-45f1-8aee-adf250553145}} .
It also explicitly maintains {{formula:52c5a80b-756a-49a8-aa26-0263ffdcebfc}} . Assuming {{formula:6263b5fb-f06c-4a4a-b215-b12782f6ed8f}} is decreasing, each call takes {{formula:dba940ad-b066-4e37-8cb9-679aeb5e80de}} amortized time.
Let {{formula:baa65b9d-9e56-4b24-874a-a85559c2a6a5}} be the initial matrix and {{formula:3e5141ad-775f-4001-8966-42bf8c462c67}} be the (random) update sequence. Under the assumption that there is a sequence of matrix {{formula:a76b3063-ada9-44f4-a14b-0037005c75cf}} satisfies for all {{formula:037d585b-11a8-400d-857d-cc3ece6d3cbe}}
{{formula:143b143b-bb00-470a-b866-7b0a9e2ab970}}
where {{formula:14906019-e2b2-4f85-8761-5bf8aa88c5ed}} is the {{formula:9c265e8e-4952-46c3-b89f-c4576293c30d}} -th block of {{formula:b4c3bc89-7dc2-44f8-bfdf-b8c117eb50bc}} , {{formula:af2b9b25-678c-4877-8e7c-34934bc6e5e5}} .
Then, the amortized expected time per call of Update{{formula:9601f149-dd99-497e-bc47-bfe1e8398d2d}} is
{{formula:61a0685f-e1fc-45ef-b302-f0f00b6afefd}}
Remark B.2 For our algorithm, we have {{formula:37541e62-76f0-4eac-bc89-bd8c8d6176c3}} , {{formula:51c8d1d8-dc4c-4aae-be36-a938589d8c52}} and {{formula:1461765a-e25a-41eb-b481-ccb46bc32080}} . Note that the input of {{formula:65504fed-912e-420f-abcd-9c0f8a2dd385}} {{formula:88a2a921-e991-4d69-98cb-2275d63eb7b3}} can move a lot. It is working as long as {{formula:9b8f8c78-0232-49ed-a6bb-42ca7633dba3}} is close to some {{formula:80854d2f-2274-47a5-88b7-f5a200dcc563}} that is slowly moving. In our application, our {{formula:738dc2f2-045e-421d-9c97-3d414bd49ce0}} satisfies {{formula:c1369d9d-83bb-4009-816e-48e8f2d25c8d}} deterministically. We keep it for possible future applications.
[!t]Central Path Maintenance Data Structure - Initial, Query, Move [1]
datastructure CentralPathMaintenance Theorem REF
private : members
{{formula:c8ee7d40-2db7-49ec-8f70-3673d74a8190}} Target vector, {{formula:4bf86b43-6dff-4537-910d-195325f6413d}} is {{formula:902437e3-62b6-4b25-b921-d52840e8a025}} -close to {{formula:8c198589-49e3-495c-acac-d1e0e0bb53a0}}
{{formula:98bf691b-df85-43db-b206-73e8b2772d80}} Approximate vector
{{formula:041dc838-9e59-4b0f-87b5-c7af04f4e937}} Constraints matrix
{{formula:7b43c30e-02f0-40ed-80c7-b0d1b9f298de}} Approximate Projection Matrix
{{formula:b9ecf0bc-d93c-4a84-9f7d-c596f5a327f0}} Tolerance
{{formula:b86691af-9c69-470d-95bd-71c3fbe3aa52}} Batch Size for Update ({{formula:0538e370-162f-44a8-b8af-eba52219d275}} )
{{formula:e7fae470-34c6-4549-abf8-932022a299cc}} Sketch size of one sketching matrix
{{formula:1571fc0e-bac3-495b-bb12-73e2b1ec1b17}} A list of sketching matrices
{{formula:706c6a8e-e3b4-4ccb-b534-d5933750f115}} Sketched matrices
{{formula:8f8754ce-21b9-4dc2-b68a-a678b8709045}} Implicit representation of {{formula:6c611824-4888-48d9-b1a0-91cdf30b0198}} , {{formula:b48620cd-1104-4346-a043-455dbccdd76c}}
{{formula:22fcc487-e575-453a-b010-124c317a4b90}} Implicit representation of {{formula:c4798dab-de0b-4363-9192-40e713dba335}} , {{formula:01dacd1b-dfbc-4e94-8c6a-73dad8f9805b}}
{{formula:b9548d9e-ab07-408f-8815-5e267b8dece0}} , {{formula:b50045c9-fc5f-4759-bffc-70c221c1d3c3}} Central path parameters, maintain explicitly
{{formula:6a3fe231-200c-4ff9-a7be-f837faf226df}} Randomness counter, {{formula:26b25d21-5eb6-4750-9de3-725be595cbed}}
{{formula:ae94c67b-5e96-4bd1-8f59-bcd69ccb5903}} Tracking the changes of {{formula:64c64273-c5c1-4a6e-a8ba-f805e38d99f0}}
end members
public : procedure Initialize{{formula:78dabfa9-1bab-424f-bebc-e4e7086de487}} Lemma REF
parameters will never change after initialization
{{formula:25b9073f-60c4-484b-9a5d-b47bd499fc7a}} , {{formula:efc08f85-5876-47a7-bfb3-d514381fbfd5}} , {{formula:1bf4baf5-a85b-47de-9fef-c37088778eea}} , {{formula:2a7fe251-b7fd-49ec-a9d6-04ebe0fb4871}}
parameters will still change after initialization
{{formula:d01e6d2a-7f4f-4edc-9e25-f91cf785973f}} , {{formula:7f2d3274-032b-404d-9bbe-4cfa540408e2}} , {{formula:f5862a58-90a0-47d1-9d79-a409b48ad9a2}}
Choose {{formula:6c6c61c9-1ffd-472f-9e8a-278710d02535}} to be sketching matrix, {{formula:588f6b01-a500-45c4-827d-c5ebc80f8cf3}} Lemma REF
{{formula:b4eae757-c77e-4049-aab7-8f43eed77a99}} Batch them into one matrix {{formula:3b9473a7-7166-4526-9fe1-5a1eba381c59}}
{{formula:1e2f4af8-2251-4341-b383-763cb975e045}} , {{formula:53514656-9c16-4a9b-a297-3f4c8a9ebd34}} Initialize projection matrices
{{formula:b44a56f1-5059-4177-b263-3793c2aa4d5b}} , {{formula:759cc4f4-d3b6-4d79-b53c-8ba9cd7af41e}} , {{formula:6b3de5e4-3adc-42ee-b1c2-1102bd3aab5d}} , {{formula:b07524f6-dffa-4a30-bb47-254ce46bcdba}} Initialize {{formula:637cd942-df1a-46a4-8143-628133bdfa1b}} and {{formula:ddbdee52-5744-4554-a22f-a083cd5d3a93}}
{{formula:27adf2f9-7226-4b74-bc1d-5491c3dbd7e4}} , {{formula:b74c41d8-00b5-482a-af24-246346e4d68d}}
{{formula:53d0f10c-0eb5-45af-8ce1-5c1c23666b8b}}
end procedure
public : procedure Query{{formula:d885757c-10d8-4b6a-a74c-cb0a569b8a48}} Lemma REF
{{formula:86b867ac-f4ed-4ab5-b184-a090615d1cc6}}
end procedure
end datastructure
[!t]Central Path Maintenance Data Structure - Update and PartialUpdate[1]
datastructure CentralPathMaintenance Theorem REF
public : procedure Update{{formula:36493853-fd26-43e4-bdd8-7bd6f2d753d8}} Lemma REF , {{formula:f05bb90a-dcd8-4308-9307-51854c9b855a}} is close to {{formula:76f3af58-f38f-4c7f-bf44-7eb4f9ed1923}}
{{formula:4e230843-3136-4093-9370-097f3a8eb143}} , {{formula:1ba7412b-1d1a-4bc3-b412-a5701feb72a1}}
{{formula:9b64587e-959a-46e8-be10-400853dbf9e5}} the number of indices {{formula:d1be4da3-cd14-4d18-b643-f60b6e0508ce}} such that {{formula:f9cc257c-9f3f-47d7-b3cf-2fec41e5823e}}
if {{formula:913b081a-1af5-456d-8798-9ff4674b14a4}} then
PartialUpdate({{formula:e0e5b437-0e11-4e8d-b293-b9ecc72bc5d4}} )
else FullUpdate({{formula:bc8e9cfe-8bec-4da6-bf3d-467c5d426413}} ) Algorithm
end if
procedure
private : procedure PartialUpdate{{formula:fba78b7a-0c1e-4763-98a3-a0411d9ae8d4}} Lemma REF
{{formula:11308fde-0af2-4865-af8e-4cb8b2144504}}
{{formula:4aab37dc-ca49-4e89-8192-4102d09b8fbc}}
{{formula:71aed5ef-f102-42df-8a10-ee4d1e1c4247}} only takes {{formula:d0fee39b-9d11-4515-bf17-1cda43ea1686}} time, instead of {{formula:a8d05b4a-84ea-4860-952e-18fadd25d093}}
{{formula:450f04ae-d7e0-4b36-a3fa-0e9bc4a4ab8e}}
{{formula:420e8f65-9f7d-40f2-aa9a-ddb79083d701}} , {{formula:dd596e98-0b12-4c1c-b2f7-049d64f0be65}}
{{formula:904928b9-f01e-4176-a1c7-d715e69e0b56}} , {{formula:e7a8edfe-b47e-4591-863b-8d6c16eacf3d}}
Let {{formula:14463155-3c61-456e-bbf7-67883798cf0d}} denote the blocks where {{formula:7b3c7473-b9a8-4442-a50a-4f82defcf55e}} and {{formula:37d8d0ed-78b5-4c35-9de4-e8cc895c12ea}} are different
{{formula:9806722c-b70b-41b8-8451-11739d3910e1}} , {{formula:50f6dd33-489f-4500-bcae-7dd6ef0eb521}} make sure {{formula:d6a405db-669d-4884-82fa-90a49c6f6e54}} and {{formula:17ea80d3-cdb5-4766-ad69-68428cb89cc5}} are close, similarly for {{formula:8ff146c9-ee76-4131-b22b-01ad40afa15b}} and {{formula:55dc763f-5dd3-4cc4-bd0d-267c9c7f15f0}}
end procedure
end datastructure
[!t]Central Path Maintenance Data Structure - Full Update[1]
datastructure CentralPathMaintenance Theorem REF
private : procedure FullUpdate{{formula:121f3365-29c3-49dd-9f2b-74fbb2583427}} Lemma REF
{{formula:58c55fe1-84c6-417d-bbb7-3e8fff260d06}} , {{formula:f8a4802e-9b2f-4841-ad44-101f1d0364f8}}
{{formula:cfa0a733-113c-46ec-a95f-71bbb1358dde}} the number of indices {{formula:08ea11bb-31a1-47c7-b0c0-f6005ea0ca27}} such that {{formula:2cec00d9-8926-4262-90ae-567a6613cc51}}
Let {{formula:6368bd51-1c0e-4d8d-8ea9-efbdb93b2e22}} be a sorting permutation such that {{formula:17bc5c65-bfe7-40aa-97df-25860eb2bdb9}}
while {{formula:58ea6522-0f06-47ee-8c27-b04d3fd0ee78}} and {{formula:af6119a2-5791-46c7-b0d6-db67e742db9e}}
{{formula:bd7b4f9c-1ca9-4ad2-8bbc-5243052ba16c}}
end while
{{formula:b62c0885-4f80-4268-8532-6bd0138abdef}}
Compute {{formula:f0ed876b-9f2a-47f4-b037-96fab99124a5}} via Matrix Woodbury
{{formula:eafed337-7f16-4ab8-891e-0b4ed6f7504d}} {{formula:835e3fc3-ab86-4477-b7de-f880fedf940f}} and {{formula:8e7b3081-ea7d-4f27-a3b5-a9e5daabbb40}}
{{formula:d5ea7eb1-dc6b-44fc-bc39-b5d413e36309}}
Let {{formula:4f81db15-29b1-4d7d-b869-faec5e13752b}} be the first {{formula:5c209ae1-0b70-4be2-b6e9-5833725fe7a6}} indices in the permutation
Let {{formula:9f248fb0-f95f-4ec0-bf52-f9a4efe1b4b0}} be the {{formula:5fd57c54-2a47-4a31-bd83-288a6827745d}} column-blocks from {{formula:ef13ec6d-dfbd-4286-8863-59f350cf7c03}} of {{formula:3c18f18e-d725-402f-a67b-58e4cc14d02f}}
Let {{formula:f6e95b4b-8521-4e2a-8b36-45193c22ac7e}} be the {{formula:e6f26aca-46d5-46e8-a916-aa84f53d9656}} row-blocks and column-blocks from {{formula:550cc9e3-28df-433c-9415-59880e6a66e4}} of {{formula:f29f48a4-2c1b-4c43-a81f-d21eeb59da01}} , {{formula:a473bfdc-5355-4235-bfaa-ba310cd4903a}}
{{formula:87238640-2b7e-43c2-a86c-eea129386339}} Update {{formula:f3fa2969-6339-41a8-9097-8b01299960cf}}
{{formula:c6ec1347-4966-4fe9-9790-9c556cd7ab38}} Update {{formula:12a3dfe8-c0bb-4aa3-991d-409db7dee664}}
{{formula:58ae782a-69ab-4b60-b323-8207dd634604}} , {{formula:f140d698-c049-49aa-9e58-86229a8df767}} , {{formula:9f8123ae-609c-4fc2-8722-406a09489eb9}} , {{formula:67f57fde-dd7f-4eee-8b14-1625d1808e46}} Update in memory
{{formula:23e3ef1c-01db-4fbb-9727-5984be785845}}
{{formula:55b4621b-8001-4f2a-a045-a31dee43d8af}} , {{formula:5a5f306a-9524-4219-93ae-a8f2670f6325}}
{{formula:6dfa3c5b-587e-4c31-81b6-73d414395afb}} , {{formula:4ccbca2c-d5b7-44b0-8fc4-0d550f164ee8}}
{{formula:65c34676-8e72-4252-aedf-b10563c6732e}} , {{formula:54af05dc-9af7-4391-ba1d-8886042c3e00}}
Let {{formula:7e89e43d-92e8-48fe-aec1-f3e4499f57ee}} denote the blocks where {{formula:eac0b6f8-1697-4023-8dfb-086f3a02420f}} and {{formula:4e174fc8-135c-460f-b05b-57fa63239d5d}} are different
{{formula:80c2d082-20c5-466e-929b-bb4a202989d6}} , {{formula:2b19b434-dec8-4a76-b69c-d0b8eee54cad}} make sure {{formula:664c2292-f214-4e50-a386-785a0d748447}} and {{formula:b2134c80-5767-4b9d-86b2-9f3185b99010}} are close, similarly for {{formula:5f6dc5e7-3f6f-44c6-a117-d97f991dc3b8}} and {{formula:152613f2-567d-4614-bbfb-a148504679fa}}
{{formula:1437319c-65f2-46d3-8f1d-fd7d33156cee}}
end procedure
end datastructure
[!t]Central Path Maintenance Data Structure - Multiply and Move
[1]
datastructure CentralPathMaintenance Theorem REF
public : procedure MultiplyAndMove{{formula:e39c3b49-f99a-470c-9e47-361d92753c9b}} Lemma REF
Multiply{{formula:aa08de08-507f-47f9-a14a-036f29444bd4}}
Move{{formula:3bf773ae-05a8-4162-be07-5916b557c839}}
end procedure
private : procedure Multiply {{formula:fbf99b1e-a28b-48b0-830a-74533991a41f}} Lemma REF
Let {{formula:c4d6f2e4-5cf1-4867-9790-3d314f23c4de}} be the indices {{formula:a725238d-95e3-4d4c-86b0-948d57dca428}} such that {{formula:1a35feed-d992-43f9-a1ee-f57d67dd4ecb}} is false.
{{formula:6fd75bc9-25ff-40dd-8964-aeb4e95a74f4}}
{{formula:fd83d755-bbf1-4bd4-82f4-aaff47fa0223}}
{{formula:a9eb2068-8390-46a4-b4ac-ef418c105f8a}} {{formula:ef25b480-35cc-4388-a893-136725b5e57e}}
Compute {{formula:56961382-2334-44df-a6b1-ace91af38d94}}
{{formula:099e76dd-986b-461b-9501-25d44d8e5aed}}
Compute {{formula:3cf051a2-f027-4470-af4a-2716226ad4c4}}
{{formula:02b6eb3e-7c32-4902-b0aa-27c4a55d6865}}
{{formula:20b69b90-2229-487a-8e4d-827a43ca36d8}} Increasing the randomness counter, and using the new randomness next time
Implicitly maintain {{formula:a46f5aac-622d-4543-a58f-72813046ab6a}}
{{formula:f5177218-31fb-4830-a4fe-651070e18a39}}
{{formula:2cfe1df0-ea7a-4318-a0d0-0669df336d40}} Implicitly maintain {{formula:54db54fb-7f76-4dd4-88f7-d9365c4641ad}}
{{formula:94db0ce1-02cd-414a-9a0e-054a3d7b6878}}
{{formula:88996aed-c9b0-437e-a02f-c303e7be28fa}}
end procedure
private : procedure Move{{formula:86eb5b7a-93f3-45d7-8ba3-35760350f57d}} Lemma REF
if {{formula:cb1f9f12-2037-4497-a631-9d57715424db}} or {{formula:27f178ea-db4c-490b-a183-21484e81ac8d}} Variance is large enough {{formula:9a336633-feaa-4e98-9c65-0a05487a1a5d}} , {{formula:a90dcbc3-6c4d-4bc6-99c1-aa23d288fb82}}
Initialize({{formula:f7d20ae2-4302-47af-b6dc-c7bdf267ec22}} ) Algorithm else
{{formula:a090b561-48d0-4369-8c34-1936dd55610c}} , {{formula:fad87131-4f29-4adc-a940-12eb48ecf5b5}} Update {{formula:67077db3-68ad-4a45-9618-9c20b468277c}}
end if
{{formula:cfb5119e-f74b-45fc-bcbb-01de855432ef}}
end procedure
end datastructure
Proof of Theorem REF
We follow the proof-sketch as {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}. The proof contains four parts : 1) Definition of {{formula:610edf08-5d38-43ce-9bf2-ba189f783f69}} and {{formula:988b7772-1e0d-46be-bb00-e7e53e6b1a3b}} , 2) We need to assume sorting, 3) We provide the definition of potential function, 4) We write the potential function.
Definition of matrices {{formula:7e346472-bbe0-4af5-a6d6-b1f730b2acd8}} and {{formula:6fadebb4-ac33-4bd4-bce7-804fc92d7f8e}} .
Let us consider the {{formula:52a5bc98-ba10-4bd5-92a7-15ac1d1dc2c3}} -th round of the algorithm. For all {{formula:3540e12a-92b0-4adb-bca7-fa7e47b25ba5}} , matrix {{formula:2cbd7b56-8822-4284-a764-6bfa539cc2b1}} is constructed based on procedure Update (Algorithm ) :
{{formula:7a5a0556-97c5-401d-b900-d9431e9524e3}}
and {{formula:8542c398-f5c8-4fba-9fc0-3bc2e0f00e74}} is a permutation such that {{formula:24f08c52-625f-47c3-8c3a-3a4c2d09b41d}} .
For the purpose of analysis : for all {{formula:ae22eb10-8775-4011-bd02-c1e8b60dc715}} , we define {{formula:fa68d926-fae6-48bf-b348-c7db3f582b01}} , {{formula:76a0f648-d1bb-4e13-a737-b364e1004db9}} and {{formula:058d150b-13ce-400a-b24c-50fbf2ff1aa8}} as follows:
{{formula:42f0f04a-5d3f-4011-893c-5358a2ada39a}}
where {{formula:a77e6cd8-1b42-4c58-b85b-4253d24f5a7a}} denotes {{formula:2c5d7b40-0a0f-4579-9338-dc5340a9780c}} .
It is not hard to observe the difference between {{formula:2014ff96-0a9a-4f1d-b9a0-b045e0fcd40d}} and {{formula:e89a4afd-b778-4d92-b4fe-af110a4f4ece}} is that {{formula:2705568a-e9ee-40d0-b083-c76f93485967}} is changing. We call it “{{formula:425a7fe5-bd29-40e8-b759-343b63bf885e}} move”. Similarly, the difference between {{formula:117cd1f2-c45b-4cdf-ba31-931014bd9d4e}} and {{formula:3c75c071-c8c2-4d11-8906-a4622a794822}} is that {{formula:cfa7ec68-b75d-4807-a63c-39f9f49f88d2}} is changing. We call it “{{formula:1086955d-0f85-40e3-8ae5-44bfdcf2e5e6}} move”.
For each {{formula:61755614-727e-43dc-b469-58aedaeaea62}} , we define {{formula:4f4cad7a-3a5c-4e2e-ad52-28be84248701}} as follows
{{formula:b9b839da-2f8b-40cd-8188-af6945218997}}
then one of assumption becomes
{{formula:50875949-107a-4af1-96b8-475acecda01f}}
Assume sorting for diagonal blocks.
Without loss of generality, we can assume the diagonal blocks of matrix {{formula:271bc442-e379-4dae-be9f-48270e72b319}} are sorted such that {{formula:9e4a8b5c-1800-4cf9-a94d-f420a7b3a539}} . In {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}, {{formula:c50aa21f-4a3a-449f-8905-dfb30cab6d93}} is a scalar. They sorted the sequence based on absolute value. In our situation, {{formula:778a2524-2442-4588-a3c0-f91ec333fcef}} is a matrix. We sort the sequence based on Frobenius norm. Let {{formula:26be052b-3c8a-4e82-9efe-93674bbfecbc}} permutation such that {{formula:6e1fa393-20d0-4edc-826c-afdcc1001e10}} . Let {{formula:6e82c3e3-3797-41a7-8fa0-6301e9895f16}} denote the permutation such that {{formula:758062ac-52bd-4fe7-941a-a4469c940292}} .
Definition of Potential function.
We define three functions {{formula:d57207d4-5745-4177-9d23-bccddb450a0d}} , {{formula:9281d58c-742f-40e6-af04-8c1d618069ff}} and {{formula:833be4d6-8daa-44c2-aebc-ae7fb3a12bd4}} here. The definition of {{formula:39546b48-850c-4568-9741-c6f17a8d47b8}} is different from {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}, since we need to handle matrix. The definitions of {{formula:0306ceaf-c0c1-414f-8b24-621f5bc2658d}} and {{formula:c821526d-8ec3-488b-befa-7d5bd592a867}} are the same as {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}.
For the completeness, we still provide a definition of {{formula:21511a6a-8fc2-4cd2-ae52-3fad4361c9c8}} . Let {{formula:350c5d34-7192-4bd1-a7f7-19bb2fcd2580}} be defined as
{{formula:10c57808-ac02-4a07-b3c0-4156925ade92}}
In {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}, the input of function {{formula:0dd363bc-5a7c-424f-bea3-f7d9171f8770}} has to be a number. We allow matrix here. Let {{formula:ae401b6c-423b-4061-83d8-2a16a67742fb}} : square matrix {{formula:47a8ce32-6b6b-4ccc-9246-b07186e2368c}} be defined by
{{formula:8a6e4f10-c6a8-4ab0-923b-ce8ff468f0d0}}
where {{formula:6b2f0bce-cf72-4119-9aed-ac51d1adf4a0}} denotes the Frobenius norm of square matrix {{formula:c378144b-5399-4fd1-bfdc-886ab8d2684a}} , and let {{formula:7ca598f7-d73b-48c5-adb1-5c8dee559488}} , {{formula:36ba6e53-6601-430f-b185-40cc076a3367}} where {{formula:84af2686-8ebd-4884-bc6b-3e02dd943447}} is the vectorization of matrix {{formula:769a581b-42d0-4bda-bcbf-3e7bd12177bc}} .
For the completeness, we define the potential at the {{formula:24e78db4-ee40-4655-8cf5-8751bdadd0ba}} -th round by
{{formula:b7c829a8-4ccd-40d1-acc6-b473736fce23}}
where {{formula:29cbf4ce-de94-4ea8-b624-0aab219523dd}} is the permutation such that {{formula:e72d58fd-034f-404c-91cc-39182d681d5d}} . (Note that in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}} {{formula:1ce6baf0-9ef9-4fa6-95b7-e897e9e7ff07}} should be {{formula:3b4e99b0-1e39-4b85-b2e9-c82f332951bf}} .)
Rewriting the potential, and bounding it.
Following the ideas in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}, we can rewrite {{formula:0f8f1f9b-0cf3-469b-995d-26aad10851e7}} into two terms: the first term is {{formula:2b1fe2f0-1140-4b9a-80e9-b00b7e9f4ef9}} move, and the second term is {{formula:d2d643ee-7ee4-40bf-83d7-914ae73c1d26}} move. For the completeness, we still provide a proof.
{{formula:855f8f95-b9bd-4d90-aa90-4057a8a29a49}}
Using Lemma REF , we can bound the first term. Using Lemma REF , we can bound the second term.
Initialization time, update time, query time, move time, multiply time
Remark B.3 In terms of implementing this data-structure, we only need three operations Initialize, Update, and Query. However, in order to make the proof more understoodable, we split Update into many operations : FullUpdate, PartialUpdate, Multiply and Move. We give a list of operations in Table REF .
Lemma B.4 (Initialization)
The initialization time of data-structure CentralPathMaintenance (Algorithm ) is {{formula:597bd372-a09a-4875-bccd-a0176fa7ca57}} .
The running time is mainly dominated by two parts, the first part is computing {{formula:02f4bcd6-fcdc-4ad8-bc57-1a522762d050}} , this takes {{formula:5419f92f-ccb0-4f69-9405-14662f1b2b9b}} time.
The second part is computing {{formula:cd651476-e015-473b-a462-5bf7937bbffb}} . This takes {{formula:30836e35-597b-4f66-aeee-3714f2628e16}} time.
Lemma B.5 (Update time)
The update time of data-structure CentralPathMaintenance (Algorithm ) is {{formula:db31a6df-84cc-47f0-a771-5c93fafabf9b}} where {{formula:74ad4850-8263-48dd-ab0d-06aa1946bae6}} is the number of indices we updated in {{formula:620f45b6-4694-4085-9e26-fc1f07f51566}} .
It is trivially follows from combining Lemma REF and Lemma REF .
Lemma B.6 (Partial Update time)
The partial update time of data-structure CentralPathMaintenance (Algorithm ) is {{formula:5d0ee74d-c1bf-4d05-9952-e254328ce959}} .
We first analyze the running time of {{formula:bf1087f6-f2d5-450c-a644-c060f4a25ff1}} update, the update equation of {{formula:a967cdbb-f5a3-4dc7-a721-1fc522bfd1cc}} in algorithm is
{{formula:d65705c7-d175-4750-83cb-de0d82998066}}
which can be implemented as
{{formula:14752e66-9865-408e-a570-580b3d15f3b7}}
where we only need to change {{formula:fff790c4-6989-4235-9145-86419fab1d59}} row-blocks of {{formula:6beb1116-7ca1-4b37-86f7-409e1e6979c9}} . It takes {{formula:de4f6bc7-17ee-4f4b-8bdd-2094da0f715d}} time.
Similarly, for the update time of {{formula:ea89dc8a-f236-4157-9f88-251b5088f331}} .
Next we analyze the update time of {{formula:8eb64bb4-2d9c-4530-8bb7-93640adf2ed0}} , the update equation of {{formula:73a629c3-debc-4f56-aa86-e5bdf863c0d7}} is
{{formula:9d1333d0-7275-4ffd-b318-5366e54850ea}}
Note that the difference between {{formula:9205b80e-7291-4465-9315-a7a9f65e1f16}} and {{formula:b837df86-db18-415d-9936-bd8481594c52}} is only {{formula:3dcbbbb0-4911-4f7c-a8e6-3beb07f84216}} row-blocks, thus it takes {{formula:d3e899cc-6305-4d87-8657-a821796dc3bd}} time to update.
Finally we analyze the update time of {{formula:e8cbff38-37fb-4acf-acf1-8ef89d3871ca}} . Let {{formula:9cd186d8-31a8-4635-b7d5-56705fa037d9}} denote the blocks where {{formula:a7e12e70-0638-4b04-84e0-7aceadd636c1}} and {{formula:2c442ce7-5409-4b26-b61a-302542e44f46}} are different.
{{formula:7f946f5b-7fba-4c31-83fb-e1f8263d4910}}
This also can be done in {{formula:9717ec09-1afb-497d-82ef-698db285ca3c}} time, since {{formula:ff92a142-0659-4080-b082-7cacd19a9eb7}} indicates only {{formula:1abe4662-9d1d-4e54-9823-09a49e6750fb}} blocks.
Therefore, the overall running time is {{formula:96e761cd-0089-432f-983a-2d981865cf99}} .
Lemma B.7 (Full Update time)
The full update time of data-structure CentralPathMaintenance (Algorithm ) is {{formula:fcd029b3-fa8e-4c99-9159-09f605a2dab1}} where {{formula:7555ea57-868b-4e95-a487-fb9da5cbdf7b}} is the number of indices we updated in {{formula:46efade8-a56c-42ec-91ad-bb9653a5c2ab}} .
The update equation we use for {{formula:303e1e54-9cea-498d-82bd-d76eca63f9ca}} is
{{formula:a4902631-73fe-4ce4-885d-003eeaadb83b}}
It can be re-written as
{{formula:c6f3943d-1ddc-47dd-8bdf-1150fd26fbda}}
The running time of computing second term is multiplying a {{formula:2bf86099-24a6-4967-8a8b-f4f919ea5716}} matrix with another {{formula:0903eeff-7bb1-40b7-877a-548c5a1efdcf}} matrix. The running time of computing third term is also dominated by multiplying a {{formula:dec7220d-52a3-49f9-8d70-005a8fd47399}} matrix with another {{formula:f5b4cce8-f04f-4e76-8b54-937271281481}} matrix.
Thus running time of processing {{formula:cdd6b4c2-6498-4bb5-b9ac-311065d4ce5b}} update is the same as the processing {{formula:50cba709-ae46-49f0-8f59-bcdc95d367a9}} update.
For the running time of other parts, it is dominated by the time of updating {{formula:f9dd427e-612f-43ab-837f-4987c47ae4ae}} and {{formula:b80a7180-5db6-4692-af95-fb942bdcb635}} .
Therefore, the rest of the proof is almost the same as Lemma 5.4 in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}, we omitted here.
Lemma B.8 (Query time)
The query time of data-structure CentralPathMaintenance (Algorithm ) is {{formula:e7a1a0f7-9bbb-4563-93f4-4dfd8d4bb450}} time.
This takes only {{formula:e6319bac-69e6-4986-9505-e39855df5adb}} time, since we stored {{formula:7e270f32-9c56-4a73-93fb-79e5023d13ed}} and {{formula:7a9d04fd-f3bc-4e08-aa2a-17144537fcbf}} .
Lemma B.9 (Move time)
The move time of data-structure CentralPathMaintenance (Algorithm ) is {{formula:b87379a6-15d2-486d-ac82-a708a6288b6e}} time in the worst case, and is {{formula:d100e429-0f27-4ea3-a2c9-6843192e66a5}} amortized cost per iteration.
In one case, it takes only {{formula:41ad30c7-d923-4ccb-942f-a7a6459a4fee}} time. For the other case, the running time is dominated by Initialize, which takes {{formula:07062c74-2749-44f1-80b3-15c218150947}} by Lemma REF .
Lemma B.10 (Multiply time)
The multiply time of data-structure CentralPathMaintenance (Algorithm ) is {{formula:8e0d0a16-1817-4532-8e75-fc8ea843f2cb}} for dense vector {{formula:dab4504f-b170-49a6-8e0a-ef2d012efaa2}} , and is {{formula:439ced66-3035-41ab-a49a-7920651d3e08}} for sparse vector {{formula:fb52e6c7-526c-4f0e-9c2d-47757806e44e}} .
We first analyze the running time of computing vector {{formula:c15e266e-f719-4059-a3e2-60f4f241eae9}} , the equation is
{{formula:40a76243-e08f-4a88-8564-9bb1ba7e913d}}
where {{formula:a80b874f-ccbc-4bca-aac8-a84f6fbce2b7}} . Let {{formula:5b006036-4eb3-4c98-ac11-eac03834472f}} where {{formula:bc932811-4b8f-48d8-8a27-e8b66628a15f}} is the number of blocks are different in {{formula:89da760d-00ab-4780-a7a5-56803695f1dd}} and {{formula:0a7f4e60-ba2c-41a1-9922-8001f1beb51c}} .
It contains several parts:
1. Computing {{formula:57f64451-a716-48d6-b915-ec2231e5fedb}} takes {{formula:067d2113-70e3-43af-9334-bf702cd7ccc8}} .
2. Computing {{formula:83671f40-ce7b-4727-bca6-deabd8b9a300}} that is the inverse of a {{formula:30f45bac-1e1d-464d-a817-579a16558a14}} matrix takes {{formula:a4d075f7-2c21-4c2e-89aa-86b0535eb081}} time.
3. Computing matrix-vector multiplication between {{formula:fe8ad85b-0309-49d3-9436-8ac84564db50}} matrix {{formula:347233c9-9e7c-4aa2-882e-3e0fdd4fea4b}} and {{formula:5c8281c6-8a48-482b-a709-67e5fa1797c8}} vector {{formula:ee79a9d2-bb92-42a7-9dff-70cdf1397dfc}} takes {{formula:95668cf0-f329-4efb-82b5-e99b3a7f009c}} time.
Thus, the running time of computing {{formula:82c13df5-5fd2-4d55-b9dd-237a458f5abb}} is
{{formula:b0709610-d203-40ff-9ff3-10538f90bd15}}
Next, we want to analyze the update equation of {{formula:b7219ac3-feea-4327-9de1-7b8a937471f9}}
{{formula:f8e976e6-6beb-4a4d-b964-d76ce7fb9d94}}
where {{formula:f8cd53c8-ab5e-498c-a55c-8504a637dc1b}} has {{formula:8b1846bc-a284-4f45-bf2a-e2c9fcfc0a62}} non-zero blocks.
It is clear that the running time is dominated by the second term in the equation. We only focus on that term.
1. Computing {{formula:7e2f8674-5b7d-4919-98f2-d0c83b1828fe}} takes {{formula:b2176931-7248-4107-920e-878fa7759073}} time, because {{formula:620b457f-598a-4d01-8bc2-ca622bbf0ed8}} .
2. Computing {{formula:db22cafd-a19a-4328-bd39-7d99b6331baf}} takes {{formula:240bce7d-903e-45c6-b9eb-b51ae1216b4b}} time. The reason is, computing {{formula:3ff6d06f-b843-40ae-9c7b-0e752d6f4a62}} takes {{formula:f124574f-8f7b-4248-8c88-e47188b653a8}} time, computing {{formula:fca5f77c-887e-49fd-8b7f-5b1ca01623e2}} takes {{formula:9ee75ed1-ff09-419f-a67a-3ba5fe152d90}} , then finally computing {{formula:e6f4ede4-cd13-4f71-bb93-c616b5931d25}} takes {{formula:135a9414-9e1f-4926-be04-1ef0eac21b0e}} .
Last, the update equation of {{formula:cddc7988-99c6-47f3-868b-a9df64104240}} only takes the {{formula:c4085c2c-69b5-4494-9664-cf5740c4289c}} time.
Finally, we note that {{formula:22a0205a-2ecf-48d7-bb4b-a0d595c22ca9}} due to the guarantee of FullUpdate and PartialUpdate.
Thus, overall the running time of the Multiply is
{{formula:b1f03ffa-f538-4632-9e79-c616dac95d6c}}
where the first step follows from {{formula:751617f6-4f7c-453e-9109-9a939c826b7a}} and {{formula:666af48d-c545-46d7-9509-ec6b3bafec55}} , and the second step follows from {{formula:1a38791b-a207-4d12-8c61-45859ab78e8f}} , and the last step follows from {{formula:888a4ed0-c6aa-42b2-acf3-998d470146f2}} .
If {{formula:44b02428-8e0a-4ea9-9a1d-05349eb2f761}} is the dense vector, then the overall time is
{{formula:a8fb79ef-d036-4788-bcbd-9dee8ce396df}}
Based on Lemma 5.5 in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}, we know that {{formula:7f5dceea-55d5-4687-84eb-aca0ba04623b}} . Thus, it becomes {{formula:88f151a8-c4cc-440f-a26b-d17171c1d8c8}} time.
If {{formula:a8e51b69-3970-4a19-bbaa-644b8f1f4a8f}} is a sparse vector, then the overall time is
{{formula:47b19ecd-9983-44a2-8e02-0ef1b9680a12}}
Lemma B.11 (MultiplyMove)
The running time of MultiplyMove (Algorithm REF ) is the Multiply time plus Move time.
Bounding {{formula:f7262c21-782d-4313-b95d-fbbc6001c6ad}} move
The goal of this section is to analyze the movement of {{formula:a258a2e7-08bd-413f-8e88-d54791ac14d5}} . {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}} provided a scalar version of {{formula:18196f63-894a-44ff-a250-6abd95b3b81d}} move, here we provide a matrix version.
Lemma B.12 ({{formula:b599abb9-4dc8-405d-948a-c9739822af49}} move, matrix version of Lemma 5.7 in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}})
{{formula:2b7bb5ce-5e89-475d-b0ea-d8d078d44d45}}
In scalar version, {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}} used absolute ({{formula:3e38ce82-1a55-461e-8dee-f31a862f2bf5}} ) to measure each {{formula:a55d8464-00f8-45f6-a905-b66ab02a82ab}} . In matrix version, we use Frobenius norm ({{formula:a9e16c21-26e7-4612-b30a-075748c8bcd0}} ) to measure each {{formula:51235ded-01a3-4e62-8186-3072ad06342c}} .
Let {{formula:2da7f808-d649-4c0c-be76-fa0ee4493940}} be the set of indices such that {{formula:101e59c8-0ce4-407c-958f-b1b041f2b719}} . We separate the term into two :
{{formula:7ea18752-638a-44e4-9a25-095219f3cd49}}
Case 1. Let us consider the terms from {{formula:04f2d73b-f545-4204-a46c-b080dc99242c}} .
Let {{formula:a3e9e088-0516-4c49-9ddb-d2f4e89deb99}} denote the vectorization of matrix {{formula:194f783e-97b6-48a6-8d2e-2ee69c84ba86}} . Similarly, {{formula:011a06a1-c8b1-49f8-9eb8-255051389079}} denotes the vectorization of {{formula:2c3957b1-2585-4c0e-97a8-eb3f559c99f5}} . Mean value theorem shows that
{{formula:5dc9ecef-3b25-4277-8a47-fd01c9f829bd}}
where the second step follows from definition of {{formula:1cdd386a-a6f6-4fd8-9041-ad238c083704}} (see Part 4 of Lemma REF ).
Taking conditional expectation given {{formula:a4bfa6f4-1acb-4dd5-8f55-8ebd8831c2a3}} on both sides
{{formula:87a6244f-771c-40ea-b62b-6b6d22fcecea}}
where the second step follows from definition of {{formula:38c0d05a-4afb-4b56-96ce-ffd5597bb90e}} (see Part 4 of Lemma REF ), the third step follows from {{formula:bcbd69d9-5b62-451a-b2cd-c0d4a2326758}} , and the last step follows from defining {{formula:b66b5034-ad0e-4175-94cf-149280509c81}} and {{formula:02c6de1c-38bc-4bb6-90b7-68aa74450bb0}} as follows:
{{formula:81aa47f6-24d2-4225-90a9-6d275abe2ef2}}
To upper bound {{formula:718bb443-8c66-4fb0-b7aa-8ee9ce023af5}} , we need to bound the following two terms,
{{formula:116f26a8-efc5-4f1b-bb1d-a79264cbe8df}}
For the first term (which is related to {{formula:6268628f-4410-4488-bf9a-3c1385c948e6}} ) in Eq. (REF ), we have
{{formula:76224807-711c-4ede-84dc-677f360986dd}}
where the first step follows from Cauchy-Schwarz inequality, the second step follows from {{formula:6d0ccc02-2698-4ade-86fe-a0d61846da9b}} and {{formula:24ec3479-2f96-447e-a630-1e9d58ef8260}} .
For the second term (which is related to {{formula:25ac1909-6703-47d3-91de-ffb8769d5351}} ) in Eq. (REF ), we have
{{formula:b83d31aa-3a65-4b0f-bb5a-338eea1947ef}}
Putting Eq. (REF ), Eq. (REF ) and Eq. (REF ) together, and using several facts {{formula:1d678503-a70f-4733-9c16-38054a11d1d8}} , {{formula:cbd7897d-7e7c-415a-ac68-82b7c8006cac}} (from part 4 of Lemma REF ) and {{formula:5eeba930-1a2e-4f44-8e08-d452a1cded3a}} (from Lemma REF ) gives us
{{formula:5bf5b71e-5980-4ac1-8b55-85def5568115}}
(Note that, the above Equation is the same as {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}.)
Case 2. Let us consider the terms from {{formula:7863c202-adc2-47b6-99f9-6a326909c206}} .
For each {{formula:490cec0a-e75f-420c-907d-2ede349c54c8}} , we know {{formula:83a01598-6fdc-423d-97cb-7bfdbb64f5f3}} . We observe that {{formula:3cf4adf6-cf43-461e-8421-2cf101168f02}} is constant for {{formula:fa4823df-f723-438b-8fa7-702129e72459}} , where {{formula:a65377ba-e01d-4bf0-8441-60a82f1fd069}} .
If {{formula:ff3ede97-0937-4aaa-9587-75581e90b9e0}} , then {{formula:039d849d-9756-4bf3-9d1f-ee4899d2d195}} . Therefore, we only need to focus on the {{formula:d771ad9f-ace1-4999-bb4b-fcfabd1eea3d}} such that {{formula:9e45b0fb-e5a1-470d-863b-6d2ed1db831f}} .
For each {{formula:799bdf9c-841a-4ef8-8798-f2f4780f3ba9}} with {{formula:eeecff4b-1f74-40d0-839e-ee4316bf6401}} , we have
{{formula:5e4fac28-3878-4ccf-a0d7-84fedf13d46d}}
where the last step follows from {{formula:3a6fe540-36b5-4258-8eb0-2a86072177c8}} .
It is obvious that Eq. (REF ) implies
{{formula:511da144-7447-454f-b09e-97ff68b7677c}}
But this is impossible, since we assume it is {{formula:a0f69c4e-3e89-40bb-aedf-1b1dbbae5419}} .
Thus, we have
{{formula:5de5d946-f2d9-430c-9b5f-7ebc70d20062}}
We state a Lemma that was proved in previous work {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}.
Lemma B.13 (Lemma 5.8 in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}})
{{formula:56a54744-a025-45cd-8c2a-21de18dbe57b}}
Bounding {{formula:960104cb-d0b0-4112-93ae-2f5e69cab642}} move
In previous work, {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}} only handled the movement of {{formula:5e4f9435-a665-4f43-b421-6f1d090666ee}} in scalar version. Here, the goal of is to understand the movement of {{formula:42661241-7079-42f8-b3a8-b0dfd4742a67}} in matrix version. We start to give some definitions about block diagonal matrices.
Definition B.14 We define block diagonal matrices {{formula:7c029165-a03e-4114-9b68-feb7fe397d97}} , {{formula:9d0231f7-041f-4ef2-91c3-b7e68f0468c0}} , {{formula:774caf56-b761-4615-9795-8589e9b28d87}} and {{formula:6bb40b1a-1cff-4cad-8618-f0ce9b9f8fd1}} as follows
{{formula:f9a2440a-409b-4bed-a7ad-59882687a72c}}
Let {{formula:ea8af0e2-4809-4269-a483-8461ec9f6421}} denote the error between {{formula:ffaa61b9-fe0f-4f37-9484-732547604e36}} and {{formula:5c1904f8-1024-4437-8dc0-023017ad7e6b}}
{{formula:a062d4be-2b61-49e8-8058-aecf088cd05a}}
Lemma B.15 ({{formula:670ad223-8590-4231-aebd-d20b3fcc48be}} move, matrix version of Lemma 5.9 in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}})
We have,
{{formula:2242c488-d981-4fb3-96dd-4d5102edd218}}
To prove the Lemma, similarly as {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}}, we will split the proof into two cases.
Before getting into the details of each case, let us first understand several simple facts which are useful in the later proof. Note that from the definition of the algorithm, we only change the block if {{formula:7cb5940d-7e0c-4911-a76c-d7b5460ef7f6}} is larger than the error between {{formula:afd301d0-4264-4b23-b690-d5373ac3c3cd}} and {{formula:e4549983-1bc3-4ee9-b0e2-fe3d3aaa2ad3}} . Hence, all the changes only decreases the norm, namely {{formula:e625c053-756c-4fe4-94d6-6d299d1b26ca}} for all {{formula:03f8d7db-6d84-4f47-866d-0c4d8143945b}} . So is their sorted version {{formula:520d9cc8-e502-461f-b3eb-04eb5d974f7d}} for all {{formula:eca96d2f-b22f-4582-babb-3176a8738fe7}} .
Case 1.
The procedure exits the while loop when {{formula:b85abe58-8c8d-4596-b7c7-a280e9868d93}} .
Let {{formula:5e17678d-5603-48ce-9179-3d9585da090c}} denote the largest {{formula:59f60913-e748-4c70-a306-5e1758f455a7}} such that {{formula:f963b401-4cb1-43d9-b5c5-7c27659cc89a}} .
If {{formula:824e170f-ea56-4b2f-abf7-546ce8310b97}} , we have that
{{formula:7b606058-ebb4-4313-bafd-d578be62814d}}
If {{formula:cc28386d-c797-430f-855a-39d729534981}} , using the condition of the loop, we have that
{{formula:230a9fd9-2ea7-4b12-9dd5-2cfd29b52b28}}
where the last step follows from {{formula:66d52b43-e057-415c-a562-262db71f32d0}} .
Recall the definition of {{formula:047c390a-dbbb-4e15-9616-9f097da51e2d}} . We can lower bound the {{formula:724121c8-eef9-4974-a622-2dc9d1097c8d}} in the Lemma statement in the following sense,
{{formula:a77e6dc0-4495-4500-9640-f8bc595f4601}}
where the second step follows from {{formula:4f0184df-bb57-4540-8846-d488f8df2e49}} for all {{formula:fed26fe0-7b7e-4eda-a0ba-4cec58dc84a1}} .
Case 2.
The procedure exits the while loop when {{formula:91eda42f-42f0-4835-99f1-fe5d36458769}} and {{formula:a20ae9ae-92c8-45a8-bed7-b0d8b631aa82}} .
Using the same argument as Case 1, we have
{{formula:f6de861b-3717-4ed0-8d5b-10d650e42706}}
Using Part 3 of Lemma REF and the following fact
{{formula:ae4abe21-8015-4e9b-b2c7-04c6d3ce5a7d}}
we can show that
{{formula:78d38c10-e934-4a93-973b-c060db8cf683}}
Now the question is, how to relax {{formula:ab0557cf-5fe2-43f0-9e71-f19bf13924dc}} to {{formula:4fce9696-8dd2-48c3-9e86-eec2f977a5fb}} and how to relax {{formula:ce55e34a-d9bf-4d68-b6c7-b94873d32bbd}} to {{formula:ee14708c-095f-43bc-9fc7-5ad64738fd08}}
Note that {{formula:60e692b4-e204-466d-9764-bc11dc9fcc1d}} for all {{formula:e6d88728-785a-4db0-9243-cee025ed41d3}} . Hence, we have {{formula:2ce6a8c7-d038-4245-ae2c-a687287b67e1}} for all {{formula:66766f6a-a26a-4731-a034-a7cec17329fb}} .
Recall the definition of {{formula:f011e86c-fa16-4d56-94a6-02ece12f335c}} , {{formula:19246fb8-28f7-415f-9cdb-57de14575e40}} , {{formula:900c8f90-e1bf-4bdc-b6f0-ff8fe783379d}} and {{formula:0d9a112e-0503-416b-8791-12dd5ddabc28}} ,
{{formula:1b156a1b-fc7c-42ad-a5b5-222c2192f44a}}
and {{formula:68648e13-04be-47c1-89f3-4307c338a265}} and {{formula:49300075-b8d6-4a39-a3c4-239688732097}} denote the permutations such that {{formula:12252c5b-6b9a-49e5-b95c-46e7936af344}} and {{formula:44bda8c7-ba15-4074-b483-f1052864f5fd}} .
Using Fact REF and {{formula:a089d56a-6295-4088-8952-62b6e10e8be1}} when the matrix has constant dimension
{{formula:67957391-296e-43d8-b537-83ba4a512461}}
where {{formula:00df654c-fcce-4bc3-bf60-71ad33ba8fe8}} is the error between {{formula:bb782594-2b8d-45dd-becc-597713c5f9ce}} and {{formula:6c6bd084-a7e2-4392-a44a-060af5961f4c}} .
Next, {{formula:6c09cc73-707e-40d9-aaea-322e36b99dfa}} , we have
{{formula:64b7e286-2bb6-4f44-8efd-7ca049708a12}}
Next, we note that all the blocks the algorithm updated must lies in the range {{formula:9fd29deb-0732-48cd-ade1-5d820632ccf9}} . After the update, the error of {{formula:17123137-04a8-414e-bf1f-fda3f9c13f54}} of these block becomes so small that its rank will much higher than {{formula:ef6d1225-cf10-49cf-bf41-cc89d12559d5}} . Hence, {{formula:9d631add-2dfe-44dc-a8f1-5c8ccb7fb205}} of the unchanged blocks in the range {{formula:ac5f4cd5-9f68-4c20-b096-679a090a3dd9}} will move earlier in the rank. Therefore, the {{formula:7bf6fd1a-fc9f-4109-a609-f3a180f851dc}} -th element in {{formula:168b3d77-d8ba-4441-b6e7-9728ec68877a}} must be larger than the {{formula:f7df8a54-24fa-4c84-8be1-1d0ec928a4e2}} -th element in {{formula:7a04db58-2448-45d3-860c-70e5eeb47488}} . In short, we have {{formula:bfbe1d2f-caff-4df4-ba31-d601e28c7546}} for all {{formula:3dec6eba-954a-4885-be19-f8f57fc1c983}} .
Putting it all together, we have
{{formula:2c40f317-a7e6-4226-8bc1-87145a304c90}}
Therefore, we complete the proof.
Fact B.16
Given two length {{formula:375202d9-cb77-470e-93e5-0db20a12daf3}} positive vectors {{formula:d61cb959-7916-4923-b963-d246a2a86c10}} . Let {{formula:cd13be5c-3784-4fc5-b3b8-03bddce91230}} be sorted such that {{formula:6a57ec2e-55e8-459e-9736-51473a4d3cad}} . Let {{formula:a3bac158-6c26-43a5-a9d7-60e61fcba954}} denote the permutation such that {{formula:2d7e4d93-7c59-4495-8046-dd1518ce7d83}} . If for all {{formula:deb685a9-06ac-40da-89c3-d12dce287415}} , {{formula:d4e9cd24-f21c-4d5b-9a6d-62ad0cf7fcb0}} . Then for all {{formula:7e9ab236-fd92-4649-b889-024985b7a7f4}} , {{formula:2f73ef74-e9b0-433b-925b-e366953670eb}} .
Case 1. {{formula:5bbb63f5-0d2b-4397-a624-574199a99b68}} . This is trivially true.
Case 2. {{formula:754165d1-a50b-4095-be51-1406c36daeb1}} .
We have
{{formula:7663f4ba-6dd0-4295-87b7-11d27d66fc15}}
Since {{formula:c2bba452-b58d-477a-ba18-169571f03c2d}} , we know that there exists a {{formula:24763069-6a9c-4824-a565-6fe079a9613f}} such that {{formula:7045dad4-c2c3-4860-9b43-cc6e466e7854}} . Then we have
{{formula:69ece0e9-3684-45df-9ce2-bb84f99878d7}}
Combining the above two inequalities, we have {{formula:fc6b1a0f-7077-4aba-a03a-5379219877d1}} .
Case 3. {{formula:42079bcd-d3de-448a-b17d-672cc475c413}} .
We have
{{formula:e766c3b0-0104-432d-8ed1-2b980b55801c}}
Since {{formula:e475a32f-92e0-44bc-a440-4951fa89d19a}} , we know that there exists {{formula:8156bf8c-6447-4097-9200-0b4f24ec15a1}} such that {{formula:b39bfd6e-889e-4068-b35c-d2be1a14b768}} . Then we have
{{formula:5a468fd1-efc3-404c-869e-34ca41d5ee4d}}
Combining the above two inequalities gives us {{formula:d9d5b782-62f2-48b7-b77b-ad85ea9cc5e7}} .
Therefore, putting all the three cases together completes the proof.
Potential function {{formula:9470e8be-d4f7-4d3d-a2bd-6e15bd1c9767}}
{{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}} used a scalar version potential function. Here, we generalize it to the matrix version.
Lemma B.17 (Matrix version of Lemma 5.10 in {{cite:d2f60c6bfbb4c5617e9b6b58011d7abc3710e808}})
Let function {{formula:e2f97cad-18f8-4c37-a72a-076aff7947c2}} (defined as Eq. (REF )) satisfies the following properties :
1. Symmetric {{formula:b3b7eda6-8fa5-4905-ba49-61a75b3568c7}} and {{formula:8c85e56f-99bd-4b79-b387-cd970d8be741}}
2. If {{formula:723fab56-273b-49de-ad19-909550f54d0f}} , then {{formula:74d4b643-da03-40bd-a3a4-c24d59840b90}}
3. {{formula:531fb2a4-229b-4ab8-9a89-c95cf74dd0e9}}
4. {{formula:4b2164bb-2954-469d-8809-15566bcb834e}} and {{formula:ddcd12b5-b44b-4b4d-abc3-1ea939a48ee5}}
5. {{formula:8a60ec10-a7b8-45f2-b9dc-d51c7f994ebc}} is a constant for {{formula:b147eee0-f5df-493e-9a64-46965d5b8685}}
Let {{formula:cea95f58-308d-4f4d-aa12-5318080ceb96}} be defined as
{{formula:c1fd2810-cd95-4bba-aac9-39916ada166c}}
We can see that
{{formula:91735d7f-8d8d-467c-925b-85d2456378b9}}
It implies that {{formula:1c2424cf-3e07-4174-9c5a-3de1fe8b4496}} and {{formula:f5a25cc8-1e79-416c-b0d7-acf7e6a86a89}} . Let {{formula:39b69034-4305-4646-88ae-3f5288f60bd8}} .
Proof of Part 1,2 and 5.
These proofs are pretty standard from definition of {{formula:2f5b1acb-8c6b-46ae-9490-18f250ce1e72}} .
Proof of Part 3.
This is trivially following from definition of scalar function {{formula:9286ba66-0d92-4af0-87a6-b4f7af8d33ae}} .
Proof of Part 4.
By chain rule, we have
{{formula:1dec112f-5db7-4a63-b90b-70a1acddbe23}}
where {{formula:80ad49d7-2337-44b7-9e4c-c027002d0c9f}} is the vectorization of matrix {{formula:48f4fd58-0642-405f-98d1-c10230280e6f}} and {{formula:69135b63-036f-438a-a907-8da3a2c3d58a}} is the vectorization of matrix {{formula:bd788358-01e4-4123-b72e-cb8a5cdeb568}} . We can upper bound
{{formula:63204a17-29e5-49e7-9256-7ccd3b4bfb1e}}
Then, we have
{{formula:83420fc0-2749-4ad1-b2f0-567a7b44dfc7}}
It implies that {{formula:c51169e2-4b97-454a-9874-6a5bfa53563c}} , {{formula:4cb8d6f1-42db-41b7-8ef0-423a96078ff8}} .
By case analysis, we have
{{formula:d6103dab-f476-4332-96e9-d00a5ba2379f}}
We can also upper bound
{{formula:050e4897-5847-437d-a3b1-514f6385d314}}
{{formula:7ee1d78e-9009-42e9-86f5-b2796b24e720}} and {{formula:091dae22-5f23-4b06-a7f5-4016410d052c}} are close
Lemma B.18 ({{formula:d71751ea-bea1-4c75-8c00-5c0073ecb5a4}} and {{formula:a4eb0d4f-c111-4b6c-ba85-8ac2ee6a22a9}} are close in term of {{formula:7abc412b-cd63-46f2-87c6-2c95bd66efbf}} )
With probability {{formula:5cccfbca-5754-4136-8d6d-520d7822bc51}} over the randomness of sketching matrix {{formula:ec348a47-fd99-4093-839d-95df580542fe}} , we have
{{formula:b4f66479-4627-4341-8bd0-f2ba0388a664}}
{{formula:5d40514a-a5af-42ed-882b-83871d55fcee}} , {{formula:5e2190d7-08ae-42e5-9133-32a41bccb7ca}} is the size of sketching matrix.
Recall the definition of {{formula:8f705b99-78d3-4741-a05e-69e10cb834b7}} and {{formula:ad28e39a-e214-42b3-8357-543193a179e5}} , we have
{{formula:70279423-aaca-41b3-924c-4339611dd3c2}}
For iteration {{formula:7b20c40a-78fb-4cb9-a6b7-9516061260e6}} , the definition should be
{{formula:06b6f1a5-227e-4ecf-a97a-91ee621ebab6}}
For any {{formula:94ef6791-d031-4178-9501-e7bf2babe434}} , let {{formula:ec332d6f-aab8-4905-b558-5355ec6c0302}} be the current iteration, {{formula:24253fbd-b834-4a99-ab41-26ca8a51cdfd}} be the last when we changed the {{formula:5271c659-29ea-4ebb-b627-05813900f19d}} . Then, we have that
{{formula:6e0b1adc-27b3-4f36-a541-44e0da96f8d6}}
because we have {{formula:ecc8b962-7f12-42d9-8573-1cb726e23d80}} (guaranteed by our algorithm). Since {{formula:4de993d5-2d56-4570-9018-ad8ec6c58174}} did not change during iteration {{formula:e444f5e4-9b2a-4106-a34e-c3409b1622e3}} to {{formula:c4dc9200-193f-4ecb-be01-1981c9032138}} for the block {{formula:4fb7fa38-10e7-4ce0-b0c7-73c62a8bd2ad}} . (However, the whole other parts of matrix {{formula:f33e17be-92a1-4fac-9154-f199fc865a1b}} could change). We consider
{{formula:9ab2bb62-90e6-4806-aa0d-7aa5a6a8afaa}}
We consider block {{formula:e54eee69-5243-4c65-b2ff-4ecb29376e35}} and a coordinate {{formula:589bbab0-7441-4017-8add-885780f78fc2}} block {{formula:7213a1c4-acc4-443c-a484-9189247c5db5}} . We define random vector {{formula:b75d7978-617a-4447-a632-8a05bcaf8041}} as follows:
{{formula:23d233a8-9159-4e4b-9d66-063f822cb2ff}}
Let {{formula:273e21a7-f4d9-4143-8cb5-532eda309ce1}} denote the {{formula:484bbc2b-423b-46de-86ba-b889fdf183cc}} -th coordinate of {{formula:3de0e22b-4156-49a3-b304-291ebbbb75fd}} , for each {{formula:66c03d61-26f9-4a4c-b978-6d8355b1e48d}} .
By Lemma REF in Section , we have for each {{formula:fe4ea619-e847-4045-bf09-4b7d8b860a9c}} ,
{{formula:f633c539-c77b-4058-8b3b-030cc5d5d07e}}
and with probability {{formula:2ba3a9ad-f1c0-45c3-8bc5-6a5fd27298d9}} ,
{{formula:32a6668a-96c5-4bc6-83e8-95b5dfa70fe8}}
Now, we apply Bernstein inequality (Lemma REF ),
{{formula:eff93731-fae2-4e54-bb05-cd6ea1c625e1}}
Choosing {{formula:d557cdf3-0d53-48fe-a92a-70c23c3ab33b}}
{{formula:74a373b2-0afa-4929-9705-f811c472578e}}
Now, taking a union, we have
{{formula:4c5d5750-161d-4f4b-9b13-0c5dffd45b2a}}
where we use that {{formula:7609a604-ae7b-4b13-924c-2a50e9e9685d}} , {{formula:33d89def-35a0-4e8d-9c24-bbfed7dfc7a1}} .
Finally, we use the fact that the algorithm reset {{formula:b74c2c19-4a3d-4aff-adc7-df5dab502078}} , {{formula:5e4bd2f5-dbee-44fb-9c56-bd3df7bcb1ed}} in less than {{formula:261a9db7-026d-47e4-be6d-70f2362da872}} iterations.
{{formula:609aa075-864e-40de-b8fa-843e701adce0}} and {{formula:8502aa9a-f3e1-405c-937d-211049714ba6}} are close
Lemma B.19 ({{formula:96c5b492-6304-463d-a39e-fd6d66ce5ce0}} and {{formula:b3a9efd9-bf1a-4c5a-9452-b0481d51900f}} are close)
With probability {{formula:284af5a0-6ca5-4465-bfe1-2bd02b46edd6}} over the randomness of sketching matrix {{formula:2fb7f86f-0955-49c0-b331-289423c3e809}} , we have
{{formula:0c960a62-d51b-49ee-8a8b-4e59c4390218}}
{{formula:aebc4594-c263-47ed-888b-615fe47d9614}} , and {{formula:136948fb-24f5-41ba-b707-98e121e89677}} is the size of sketching matrix.
Recall the definition of {{formula:0857126e-9427-448c-bcb2-6fc2e6a246dc}} , {{formula:cc911e56-cf16-474c-8d10-c9aa7380126d}} , we have
{{formula:f8708180-5dbb-4b55-8acc-4aeb60e2ee19}}
The rest of the proof is identical to Lemma REF except we use also the fact we make {{formula:24fd9a2b-0583-4ed5-b6eb-6d0c6b0844a7}} whenever our {{formula:242ee2ed-02c3-44e8-bb63-d1ccc67a6db0}} changed by a constant factor. We omitted the details here.
Data structure is maintaining {{formula:d0c7a6c5-9575-4153-93c9-fe0ef1dffd53}} implicitly over all the iterations
Lemma B.20 Over all the iterations, {{formula:ebafc54b-f14c-4071-a1f1-237b72bd90ac}} is always maintaining {{formula:cc6cbe4c-b5af-4092-8a25-ed97c1568da3}} implicitly, {{formula:38b8d6db-55df-496d-9ca9-609addaec640}} is always maintaining {{formula:587198e6-81e3-4fd7-9dbe-b62cdd4c414e}} implicitly.
We only focus on the PartialUpdate. The FullUpdate is trivial, we ignore the proof.
For {{formula:77fa4b35-9a86-459b-b639-97a1ff40f472}} .
Note that {{formula:d1a127b5-72b4-4f61-9b22-04058c153f92}} is not changing. Let's assume that {{formula:8292ec6c-d2aa-4eff-a395-04ecad099b0c}} , we want to show that
{{formula:aded5395-0e09-4bae-99b8-4798ce3fc10a}}
which is equivalent to prove
{{formula:bddfee51-cf74-4e15-8527-e55b4037d33c}}
Let {{formula:e1fe62de-69ef-41ec-97c8-4cb035cd49ed}} be the change of {{formula:1665c5bf-6952-4351-8b1f-90bf05af2f9f}} over iteration {{formula:90b72e67-77c2-4e51-80de-c8bac524762d}} , then
{{formula:53461e43-4397-4c12-98f8-8afb795a9a87}}
Let {{formula:7a252f4d-9a5a-47e6-ad3b-b4802cc40084}} be the change of {{formula:ffcbd2f8-57dd-4888-813e-a333242128ac}} over iteration {{formula:9f9f49a9-684a-4fdc-8eed-d67915745422}} , then
{{formula:229b382c-16f0-4ddd-9bd5-8e8924f0aae5}}
By definition of {{formula:9ff74858-eb40-4447-86f2-43d8640a5673}} at iteration {{formula:a0f4fcd5-a3c8-4641-a6b9-d3ef8392ba10}} , we have
{{formula:4e23b534-f2ac-43e0-9107-e17a56c8f688}}
We can compute
{{formula:bef76fe7-d490-4e02-9fae-d022105e7ec5}}
where we used {{formula:6774c6e3-df15-4509-9109-be2a1eedf471}} in the last step.
For {{formula:0d696128-04f8-4bf3-8181-616e7593980d}} .
We have
{{formula:8a708377-268f-422c-a9e8-cb85bc7c966f}}
Let {{formula:641be02f-96e3-42f7-b1e0-a97a3ce595c7}} be the change of {{formula:4a163820-e473-4d1f-8440-e039bf06691c}} over iteration {{formula:95e20c3b-129c-4de9-817c-4d638767e90e}} , then
{{formula:c938cabe-b056-4206-9ac8-2a4153103586}}
Let {{formula:8e06d28c-b30c-4d29-ae1a-b78ff169f439}} be the change of {{formula:ae83281d-a90c-4ee6-9208-aa6b1f8b4ae7}} over iteration {{formula:e1b1aeec-2df5-454d-9e65-721915494804}} , then
{{formula:cee2c765-f90b-4a68-826a-e4ea7ec89c08}}
By definition of {{formula:b8dce976-2b82-4e4a-8544-468de21a7887}} in iteration {{formula:55d0df49-fd96-4373-976d-8a760bd433a9}} ,
{{formula:37cdc8a3-5680-4ca5-aa98-a684647f0b20}}
We can compute
{{formula:b5806641-5bd5-4606-92be-837a0da148a0}}
where the last step follows by definition of {{formula:82f71f0f-266e-47bc-99a3-cfa6a41a238d}} .
Combining Robust Central Path with Data Structure
The goal of this section is to combine Section and Section .
{{table:66b4d510-54da-42b1-b02b-1581b898b683}}Guarantee for {{formula:8f75639d-76ac-4912-9897-74d0faf9c53c}} matrices
Lemma C.1 (Guarantee of a sequence of {{formula:7f9af3ff-ed23-43fd-8ed7-7ef34d16554a}} )
Let {{formula:a7a18839-a1dc-4e11-9833-2c56d01166a1}} . Let {{formula:29bce8f0-251c-40c1-b9fd-4a7d2cc27872}} and {{formula:c9f721a3-ea03-4053-8619-983a00b20b17}} . Then we have
{{formula:381b2ea4-51f0-4acd-a2a6-23a550222018}}
where {{formula:efd9714c-ede9-417a-a980-7af45bcfd745}} and {{formula:933bf652-b195-4a69-92f1-f6ed50b9fba9}} .
For each {{formula:66faab22-d0d0-44d2-8674-8b45e8acfbb6}} , we have
{{formula:d760e8b1-200f-44e5-b087-3bd7c6dbc712}}
where the second step follows by Theorem REF .
In our problem, we assume that {{formula:2a5a5c77-3d73-4e4a-8787-7341dde435ea}} . It remains to bound
{{formula:510ae356-1920-4e1d-888d-2b717470a5d8}}
where the last step follows from definition {{formula:28a8f92e-f01f-47ae-bb62-dad50457f1d3}} .
Then, we have
{{formula:a42e26a4-546e-4653-aa9e-9c8ba1167dc6}}
where the last step follows by Lemma REF .
Lemma C.2 (Accuracy of {{formula:b9774c33-7b34-4ea0-b80a-92b35a436e6a}} )
Let {{formula:b7cfb433-ae87-45bf-adb7-0a1cc768a1f8}} and {{formula:39f1392f-7b30-493a-bdd0-8f3b6cb5e838}} be the vectors maintained by data-structure StochasticProjectionMaintenance. Let {{formula:73c4a37d-8371-4469-baf0-cc942668b314}} and {{formula:e7756b0a-a8a8-4a0f-a853-4cdb46d707e4}} . Then we have
{{formula:3542e9cb-d5bf-4bfb-b6bc-cf086b7b1399}}
where {{formula:a562e077-0da6-488f-b56b-f3a99e11b46f}} , {{formula:345061cd-a88c-4ac5-8d09-24554a879440}} is the size of sketching matrix.
By similar calculation, we have
{{formula:41ff9022-6057-453e-81e5-32f7e85498ba}}
Then, using Lemma REF with {{formula:b759923a-f789-4b87-be2e-3e0b3a5527e2}}
{{formula:000cd557-47d1-46fb-8252-524ab0bfa0b7}}
[t]Robust Central Path
[1]
CentralPathStep{{formula:abeedeb0-31d0-4514-951c-8dc5be1e5931}}
for {{formula:d24ae09b-d518-485f-a326-0a11d0384946}} do Figure out direction {{formula:768b5b15-2212-4af6-9ee6-b63f7de62639}}
{{formula:1dfb3feb-a1e4-462d-ad84-6c63353de076}} According to Eq. (REF )
{{formula:33fecac9-4223-459c-a500-7d7fa3f28ec6}} According to Eq. ()
{{formula:8d3d3898-23d2-482e-90c1-7121b644282d}} if {{formula:7432f997-9ce3-438d-bd4f-a430e975fd42}} and {{formula:a880e0a1-55af-4e3b-9ae6-16643619b658}} otherwise According to Eq. (REF )
{{formula:1eb03262-31e3-4a67-8bc0-d380a908c041}} According to Eq. (REF )
end for
{{formula:e997def5-2a0b-4595-9453-113f5579b0c4}} Computing block-diagonal matrix {{formula:68345f58-a799-4069-8c3d-485f9ef1318d}}
{{formula:cd4916c7-6904-4325-bc19-0c111d2f33a6}}
RobustCentralPath{{formula:45dae739-799d-48ba-a664-e7e8af44b70c}} Lemma REF
Standing at {{formula:d135145b-67db-4a85-90f7-7fe018ce8d79}} implicitly via data-structure
Standing at {{formula:4a52f757-e86a-47a6-a59c-33f7ff0957f9}} explicitly via data-structure
{{formula:337be2f6-3cff-4ae4-a9ec-0cb6a6d54a4f}} Algorithm , Lemma REF
{{formula:e3bbc7d0-93f5-4159-9192-83ea07ea4dfc}}
{{formula:f22ee5ae-fcfc-4b19-826e-660485537f06}} Algorithm , Lemma REF
{{formula:b52fe183-adbb-4d16-9437-84d79b69e05c}} Algorithm , Lemma REF , Lemma REF
{{formula:8b968e9f-a0ce-40ba-bb0e-cd7a332b9ae2}} , {{formula:50b9681e-98f4-4edd-a8dd-85058578b39d}} , achieved by data-structure implicitly
{{formula:b395b3aa-99e9-49f3-a5a9-c31b78cd7be9}} , {{formula:b0e79080-e235-4fb8-97f1-2e05b8bc142c}} , achieved by data-structure explicitly
If {{formula:3f03e20e-8d6a-40a6-8ab7-19c806b225b5}} is far from {{formula:cce94b9d-a3da-46fb-8953-8081d233c12a}} , then {{formula:ad5f8041-e21a-4f5c-a46e-f9ce21b7395e}}
[!h]Our main algorithm (More detailed version of RobustIPM in Section )
[1]
Main{{formula:1f093af4-54d8-444c-90b0-733c8bf10973}} Theorem REF , Theorem REF
{{formula:38f15d53-a7e6-4cbd-94f7-0a3f2d56af49}} , {{formula:fad15fa6-c872-4277-9a47-dd6e105ab2a1}} , {{formula:569ec6cf-2dbb-4672-9feb-7d6e09c93b56}}
{{formula:4e14500b-c77a-4990-9cd8-3f995390f844}} Choose the target accuracy
{{formula:a08d523d-c55b-43ad-9596-35ce89e4a8cc}} Choose the batch size
{{formula:e9f05286-1212-41f1-8102-d2287ba4d798}} Choose the size of sketching matrix
Modify the ERM({{formula:e644fe42-2a5b-43ca-8c97-a8298f4e57f3}} ) and obtain an initial {{formula:1cef2eb0-3862-4c54-907e-b3177315d725}} and {{formula:9466f24d-fbda-4c06-a258-d6c063cd2083}}
CentralPathMaintenance {{formula:d7889937-a78d-426e-90e8-8bafd46efc8f}} Algorithm , Theorem REF
{{formula:d61df274-d952-4b8b-9b0e-8c1a9b3e5935}} Algorithm , Lemma REF
{{formula:f397cf70-0a4d-4525-a8b9-ffa62dcb9f78}} {{formula:87af1a84-26f9-4bfd-865f-e0b9c7feb2e7}} are the self-concordant parameters of {{formula:40eede8d-6b2d-44a6-ad21-5a5186034844}}
{{formula:61dbf6e6-383c-45ed-b714-5caf2869d778}}
while {{formula:6b9ac88b-b224-4e1f-bf5c-144c9e31bfc8}} do
{{formula:057d9d67-e1ce-47fa-83f4-626d74177fc8}}
{{formula:c25e3410-ab64-4706-b61a-db1b13a20390}} Algorithm REF
{{formula:85969956-43ca-4bab-a683-0a72836d0143}}
end while
Return an approximate solution of the original ERM according to Section
Main result
The goal of this section is to prove our main result.
Theorem C.3 (Main result, formal version of Theorem REF )
Consider a convex problem
{{formula:d429d545-e581-4385-8c94-5fa98ae62599}}
where {{formula:bb5df0fe-91d1-478c-bfed-158cd9eaf26b}} are
compact convex set. For each {{formula:f53a775a-8250-4b9e-943e-3c2db18faba0}} , we are given a {{formula:5694aa62-263f-4720-bc50-1fa4de0e9079}} -self
concordant barrier function {{formula:ab151500-696c-4256-88ab-09500eb8fcfd}} for {{formula:f5ab6f85-6282-429d-bc7f-4aac4849a259}} . Also, we are
given {{formula:bcdc6f01-123b-4b2d-8632-36d0ab40d4ce}} . Assume that
Diameter of the set: For any {{formula:db32e69f-a1a9-4cad-96cb-4dc01a9ebce1}} , we have that {{formula:427ff762-d210-4de3-a2b8-c8d3787e1f4b}} .
Lipschitz constant of the program: {{formula:4b19e9f4-a6f3-4305-b85f-e4069e8bcb43}} .
Then, the algorithm {{formula:b0e01e97-d2ef-4943-a7d9-83f19d99142c}} finds a vector {{formula:d16eb54e-dc2e-45a3-8131-db358d5dc69b}} such
that
{{formula:206c3998-3db7-4fd3-b845-2e10023b4203}}
in time
{{formula:b5ccb96f-5792-4b36-a068-9093b801acfa}}
where {{formula:dc17814d-c2e2-4565-92a2-56d6ce8ef16a}} is the exponent of matrix multiplication {{cite:f438adfa29b0072cec3605c2cb36217248aa202c}}, {{cite:e9f448a8cc5ad7f96634107995c6f92bb60840a2}}, and {{formula:ed11c9fb-7ea0-47f7-9189-8008ba5eef1e}} is the dual exponent of matrix multiplication {{cite:45b625d473487dfdc227bafde5e2ee208bf12f38}}.
The number of iterations is
{{formula:1cdfae91-19ea-4d8c-ba6a-86985fb59e84}}
For each iteration, the amortized cost per iteration is
{{formula:6027114d-dbd2-4340-a406-5b5fced52ae1}}
where the last step follows from choice of {{formula:46b4a8cb-5bed-43e6-8d37-48c204babf07}} (see Table REF ).
Finally, we have
{{formula:b7bc4b5e-e63e-42aa-a915-dd1e50f72d7e}}
where we pick {{formula:282addde-0e47-40b1-a4e3-42281ddfed82}} and {{formula:05179a6e-cef0-4b37-b73e-702566bdc5d5}} is the dual exponent of matrix multiplication{{cite:45b625d473487dfdc227bafde5e2ee208bf12f38}}.
Thus, we complete the proof.
Corollary C.4 (Empirical risk minimization)
Given convex function {{formula:7d6d69e7-24ad-4a00-9511-a68c4ddf43ce}} . Suppose the solution {{formula:1784ef40-0bb2-4583-a3ec-d73a5dc00b9b}} lies in {{formula:0c7d14ec-1b46-4bee-bbf1-6e6da7d6b826}} -{{formula:622c474e-f243-4fb8-978d-c828315c33f8}} . Suppose {{formula:3e8046cd-bdbd-4964-b804-c53863bff18e}} is {{formula:7d27ec91-3e5e-408f-a4ce-7b01a21c60d5}} -Lipschitz in region {{formula:dcb46396-ec9c-4e49-900f-19c680863e03}} . Given a matrix {{formula:fb53203d-e518-4c9a-b2d2-b5bb42b8bd58}} with {{formula:a5a6c202-ca2a-4fad-9261-08e415ca4d1c}} and {{formula:6247de27-efa2-4750-bc52-edec1447a811}} has no redundant constraints, and a vector {{formula:f54a239f-fcb3-4f34-98d9-68616da1e258}} with {{formula:399cb395-b997-406c-b9c9-41cca6895936}} . We can find {{formula:3df54200-5195-4549-803f-bb0448a7b45c}} s.t.
{{formula:6f024855-8970-488c-8dab-b1a6cadca28a}}
in time
{{formula:a1289002-351a-4c14-8f03-d5d35e75093b}}
where {{formula:f265863e-7aa1-45f4-baa8-7b5e564413df}} is the exponent of matrix multiplication {{cite:f438adfa29b0072cec3605c2cb36217248aa202c}}, {{cite:e9f448a8cc5ad7f96634107995c6f92bb60840a2}}, and {{formula:e7f47cff-131d-48db-9cc5-0269d632514a}} is the dual exponent of matrix multiplication {{cite:45b625d473487dfdc227bafde5e2ee208bf12f38}}.
It follows from applying Theorem REF on convex program (REF ) with an extra constraint {{formula:5fb2f953-41f2-48da-90a6-96f5103e143c}} lies in {{formula:16ec5dfb-da2c-4412-bea1-0be3703f74d8}} -{{formula:0939822c-6778-4dfc-b84e-e8ec33157fa5}} . Note that in program (REF ), {{formula:7977d58e-65f7-4161-9931-91fb2ebef450}} . Thus {{formula:db2f4f00-59f5-4f11-80b1-8ba3253126da}} .
Initial Point and Termination Condition
We first need some result about self concordance.
Lemma D.1 (Theorem 4.1.7, Lemma 4.2.4 in {{cite:0db784ff96a0b4ce8987584755716d6398651c8c}})
Let {{formula:25186f88-cffa-4523-97e0-c817feaedf22}} be any {{formula:d76c6024-e6ca-4d32-9237-e0943ffbc9ba}} -self-concordant barrier. Then, for any {{formula:f8b45f18-921f-438c-a153-b59304b9fa8a}} ,
we have
{{formula:658f997d-e68e-46b4-8750-0794563c2f7a}}
Let {{formula:c15ba99f-0dbb-4886-99a4-a317d1448cfc}} . For any {{formula:f8862580-cbac-4c46-913c-84868ac4d315}} such that
{{formula:10a39e6d-ffc2-4cf1-8eca-4cc2f150011f}} , we have that {{formula:ec8e1562-3909-483f-be51-6fad61f335ee}} .
{{formula:99343c99-cf9d-4aa2-a1e0-7b5597042e38}}
Lemma D.2 Consider a convex problem {{formula:2a4e5254-39a0-4e3c-b70f-2de4f7f5be34}}
where {{formula:8d8061fa-23d6-4758-9cb3-cc856e0c669f}} are compact convex set. For each {{formula:1405666c-91e7-40b5-b677-0a9f3d0dbc6f}} , we are given
a {{formula:ba5e17f7-e294-498b-997d-c26b678cef60}} -self concordant barrier function {{formula:05cd5c83-f1d2-49a2-83e9-61490d5638ae}} for {{formula:72fe1706-c68e-440b-b4c0-5e72c420329d}} .
Also, we are given {{formula:8d48f9c6-4518-4f1e-acb7-2509cc33df2f}} .
Assume that
Diameter of the set: For any {{formula:44177a9e-05c9-4221-aea5-d82a51c99e79}} , we have that {{formula:fb73e41b-1ce0-4287-a650-2ac000af1211}} .
Lipschitz constant of the program: {{formula:135acd38-6346-49d9-8259-2d4ee4a34c8c}} .
For any {{formula:bd2af0d3-7329-43fe-b574-7b6c5aa007ec}} , the modified program {{formula:d448b2c2-fad8-4cc2-8af0-56b87bdcb18a}}
with
{{formula:a3fca838-b844-420e-b613-a543681e8bc8}}
satisfies the following:
{{formula:848e885a-93dc-462c-b8cc-f51a98d23b0b}} , {{formula:758c981b-589f-400a-ab14-eededb60e544}} and {{formula:4b6f6519-e878-4551-8080-07c89dce628d}} are feasible primal dual vectors with {{formula:01357745-0d73-43dd-b9bb-d84bb62e5faf}}
where {{formula:fa8f4c95-ffe4-4ceb-b611-85a93e2f636d}} .
For any {{formula:5a96ad52-2188-4c74-803b-97c0202cfa1d}} such that {{formula:0f3ca7fb-47c3-45e3-ad03-e7741a3a78e8}}
and {{formula:a06f53df-3587-4385-9bae-ecce15016073}} ,
the vector {{formula:36ec8633-8e48-4b34-910f-f8e732a0474e}} ({{formula:4c348f3c-1a83-445c-a4de-cfe552438ab9}} is the first
{{formula:f36ae5d2-314b-4c8d-8c09-4e5743da3786}} coordinates of {{formula:13bfa929-375f-4127-a89b-f7faa2733710}} ) is an approximate solution to
the original convex program in the following sense
{{formula:cd549594-3f04-4ed2-b4d4-f8a14df51c8c}}
For the first result, straightforward calculations
show that {{formula:ff9ac26e-c4e0-48e7-8ecb-c20c822c03ec}} are feasible.
To compute {{formula:c0729408-1d81-4840-a8cb-eb1842d0dc94}} ,
note that
{{formula:d165fc75-6892-4670-96f1-74961979f18f}}
Lemma REF shows that {{formula:327e27f5-0590-4811-a027-f395a6cbf7e8}} such that
{{formula:fdb4e9d3-b986-4e6b-8092-e64236b0b526}} , we have that {{formula:6088f526-d538-4c67-9d9b-00c335dd72af}} because
{{formula:3d031b6b-95ba-4b3a-9f29-66119c6e80b7}} . Hence, for any {{formula:13e06e74-6e26-468a-ac59-6278323acdce}}
such that {{formula:f10f4c5e-fe57-4074-a98b-2efa494c6675}} , we have that {{formula:435c9009-8c83-4ae6-98d6-fbf7f2dd8308}}
and hence {{formula:a872f391-66da-4c4f-a535-3ff586aa0069}} . This implies {{formula:c9ad7a1c-c00a-45ea-a247-52d84f886676}}
for any {{formula:a65bf5fb-b0e4-4667-ab77-91ce1ed96f38}} . Hence, {{formula:4238051c-f676-4760-a310-8931abca92f4}} .
Hence, we have
{{formula:fab52d73-1895-4754-92c7-34523606471a}}
For the second result, we let {{formula:fa2411f9-1f47-4eea-85f7-d04be2208572}}
and {{formula:7ab5784b-02cf-41cb-80c7-c4abaf8b371f}} .
For any feasible {{formula:27b99d7c-1242-4974-9b1a-88a475b15f30}} in the original problem, {{formula:d19b6bf2-46de-4649-80b8-391a5d5395fc}} is a feasible in the modified problem. Therefore, we have that
{{formula:3705a9b8-4f08-462c-be2b-2e820b838983}}
Given a feasible {{formula:fd72e702-eaef-4cd2-930e-7a6869ded241}} with additive error {{formula:bf25b059-b3ed-44a9-8028-16df9e8ae7a3}} .
Write {{formula:e8998e2c-130b-4e06-9afb-3d11c6e3c51e}} for some {{formula:ec719cfc-f59d-49ff-a069-6712a30e713f}} . We can compute {{formula:a62f7825-7c94-4129-ab19-d55ab7e06edc}} which
is {{formula:5f1effa8-a8ac-4ef6-9ce9-3da589579d94}} . Then, we have
{{formula:25826bf9-f2f6-437a-925c-5bc32b7729e2}}
Hence, we can upper bound the OPT of the transformed program as follows:
{{formula:903b8576-f3f9-445c-9775-240f6c09c231}}
where the second step follows by (REF ).
For the feasibility, we have that {{formula:66ccaca2-36f3-44b0-8d3e-a4f1b92a89e9}}
because {{formula:3a802d0d-f1a1-4912-815d-9d492e0de7c4}} and that
{{formula:2bf9dd5a-8d6c-4d7b-959b-2d3a0cb297c6}} . The constraint in the new polytope
shows that
{{formula:65491fb1-2429-462b-ba49-23c724659041}}
Rewriting it, we have {{formula:bba5abdf-c3d7-479d-aac5-e28d8f6b60f8}} and
hence
{{formula:f11868ca-ee74-44b1-8603-eebab2c06fbc}}
Lemma D.3 Let {{formula:3e7e18b0-1916-43fe-af60-afbd213a20fd}}
be a {{formula:f93cd8cb-9027-4649-8760-188f20fec836}} -self-concordant barrier. Suppose we have {{formula:c62e16c1-df53-4c56-8613-4dafef01be00}}
for all {{formula:00e6e45a-64bb-4920-b170-d4d3995983b8}} , {{formula:040ebfd8-4ddc-44b4-b58d-84263374c777}} and {{formula:df4bf08d-e5d1-4f95-8521-62504230b48e}} . Suppose that {{formula:026cb5c2-10a8-4385-8e4e-7878051481b5}}
for all {{formula:2fbc8cf7-ac97-4e92-b659-872baf82d8da}} , we have that
{{formula:f6b590c4-46b2-4b37-a229-b298e1beb693}}
where {{formula:1b619796-bb32-4c3b-b04f-ac6321237bd3}} and {{formula:510d76a4-5c92-4b34-840a-12acd433963a}} .
Let {{formula:06bb7772-bce2-4e35-b12a-f1a15344c480}} for some {{formula:384dfe34-7428-4166-a568-113e20bf1449}} to be
chosen. By Lemma REF , we have that {{formula:67aa1962-4aaa-4afe-a39b-77cdaf7c507d}} .
Hence, we have {{formula:d4edbf6d-9489-40cd-89de-6ba04d3b44cb}} .
Hence, we have
{{formula:fb7b39da-91c2-4ecf-948c-6e2f384d1e8f}}
where we used Lemma REF on the second first,
{{formula:cbf75a9a-9d9e-43d3-a0f3-6f075b21697e}} on the second inequality. Hence, we have
{{formula:847f7822-b889-4eac-b704-dfe5b40c3daa}}
Using {{formula:ff1d9236-86c5-4a23-af78-18ae7f36b30e}} for all {{formula:6a86be0a-05c5-4263-b7e8-634685b4ef31}} , we have
{{formula:45feae69-22e9-4b90-b57f-792ab4d51ab9}}
Setting {{formula:815ebca4-d4d8-4a19-ac56-efb107651174}} , we have {{formula:b66c3efd-d577-4b95-a3d3-aaf282a38921}}
because the self-concordance {{formula:2bfdcf05-1111-4acb-a3fd-8d4f9d8e9ba2}} is always larger than 1.
Basic Properties of Subsampled Hadamard Transform Matrix
This section provides some standard calculations about sketching matrices, it can be found in previous literatures {{cite:5746f9e9f9e2ea1227341a2384cc2bac87be4462}}. Usually, the reason for using subsampled randomized Hadamard/Fourier transform {{cite:25988d79d9ba449c1655ddf90a05f703b034cbbe}} is multiplying the matrix with {{formula:661f206f-17ef-486a-98bc-a47320aa4ca9}} vectors only takes {{formula:37f9a3d9-7b1c-426a-95ed-e6a38ab89533}} time. Unfortunately, in our application, the best way to optimize the running is using matrix multiplication directly (without doing any fast Fourier transform {{cite:caa03d913ed8810bcf556d32314f4140aa5719cd}}, or more fancy sparse Fourier transform {{cite:5e0fee98bc47faf70f724a53d0ef6a0f7441f8cf}}, {{cite:a94828c453f30f131b0e8aba1c1f2ff2eea9f8d8}}, {{cite:4e59d71e68e547fb6f383855e9fdef91b498d3e6}}, {{cite:56e184c4d018799eac1bb5698a8c23a5dcd43cdf}}, {{cite:cc538155201ddb145d89201389ffbfacbfc7cdd0}}, {{cite:3aa016cacc7bd424ee1740106804277505e24490}}, {{cite:cfbc885ba03e891aa9ca6fd8ead11b15f018ba4b}}, {{cite:097022b8b79c280247d4d1c6c3aef5102ec39b2f}}, {{cite:5ea60b969d0b51cbcf2b8e6f9b36b54e8698677a}}, {{cite:6105a1a0eb495e19bf7346908faf638b8b6fdb18}}). In order to have an easy analysis, we still use subsampled randomized Hadamard/Fourier matrix.
Concentration inequalities
We first state a useful for concentration,
Lemma E.1 (Lemma 1 on page 1325 of {{cite:1e92dd3c37cae5b4d7494e1dc2d1f175d3c8a144}})
Let {{formula:0dafd48f-e1b3-42cc-b7cc-82f4a637030a}} be a chi-squared distributed random variable with {{formula:beb839f7-8f88-44c5-ac00-3a4f15da0e4d}} degrees of freedom. Each one has zero mean and {{formula:741ad363-8bcf-42db-8805-e2443b6f4dd5}} variance. Then
{{formula:c468954e-a3ec-4392-a885-54f46ee16953}}
Lemma E.2 (Khintchine's Inequality)
Let {{formula:4bf54771-0843-479f-99a7-bb4bc48abd79}} be i.i.d. sign random variables, and let {{formula:910334bd-8073-4dd9-b529-fb90c6046d0d}} be real numbers. Then there are constants {{formula:5f763145-252c-46ec-8fc0-62cef3976c42}} so that
{{formula:329a8686-e99f-4e3d-976e-747652057247}}
Lemma E.3 (Bernstein Inequality)
Let {{formula:1d62c913-31db-4a82-8fc6-5298f9e70e7b}} be independent zero-mean random variables. Suppose that {{formula:833afff1-4dbb-4466-9e11-506bb911da94}} almost surely, for all {{formula:6a01bc4c-5d88-4651-bebe-af644feeb2c2}} . Then, for all positive {{formula:459f4eb1-0983-4615-95d7-2f15b3d40fe2}} ,
{{formula:7a985e08-64da-4776-9cba-0f68011b08e8}}
Properties obtained by random projection
Remark E.4 The Subsampled Randomized Hadamard Transform {{cite:25988d79d9ba449c1655ddf90a05f703b034cbbe}} can be defined as {{formula:d3d2266f-074f-476a-82e0-2ccf2607b3ce}} , where {{formula:1f94ee0a-c2bb-4e38-8518-8a49032f78c6}} is an {{formula:d4e476db-ebbc-421a-9763-dd4b1a276682}} diagonal matrix with i.i.d. diagonal entries {{formula:7883e5f3-9cd1-43e1-88f4-42ba3b8ca668}} in which {{formula:426cd0ad-2495-467e-b854-2fb7018b3c9a}} with probability {{formula:270d5095-66ff-4c3d-a095-b0de7472a71a}} , and {{formula:c9fe3031-fbc8-4867-a8ae-3a17ae2301d5}} with probability {{formula:3b9fa673-d02a-4e2c-b2c3-6f7486f6ca6b}} . {{formula:47870c01-d67d-418c-9c71-1529f18f9546}} refers to the Hadamard matrix of size {{formula:fe936d0e-5eb9-4be5-9318-ef77f94e15c7}} , which we assume is a power of 2. The {{formula:1aaa5ff4-f2e5-4769-a6f1-111ef18f8f57}} matrix {{formula:13de8c87-3be7-4399-bb56-31e1eda6258c}} samples {{formula:147129b6-43ac-47ca-b81c-0d9757a1a7cc}} coordinates of {{formula:b7f9081c-e268-4c40-a913-64311365f64d}} dimensional vector uniformly at random. If we replace the definition of sketching matrix in Lemma REF by Subsampled Randomized Hadamard Transform and let {{formula:3f8ff4a5-9821-4302-966c-f2257c15f704}} , then the same proof will go through.
Lemma E.5 (Expectation, variance, absolute guarantees for sketching a fixed vector)
Let {{formula:ba556090-53fa-46c6-8903-a665fafa7381}} be a fixed vector. Let {{formula:fafa23f4-2372-4224-b713-0cab441192a1}} denote a random matrix where each entry is i.i.d. sampled from {{formula:f52cecd3-da55-498f-a584-3a474814e477}} with probability {{formula:8506efde-5c52-435d-abde-55b5281d4a6f}} and {{formula:eeb29fb6-14f0-4581-a95e-fb89277d71bf}} with probability {{formula:1ddc4fdc-fa5b-4496-8ed0-a6f3a09113c1}} . Let {{formula:afce107d-ac59-40e3-b896-574e672794df}} denote a diagonal matrix where each entry is 1 with probability {{formula:225af7f8-0950-4e4e-b7a9-6c37cc463978}} and {{formula:62ed945d-1598-4e8b-91cd-5530c3996454}} with probability {{formula:295bf788-b8d6-4c35-a695-83ffff42ea0c}} . Let {{formula:f8a1418c-cade-4002-aa9e-a024781e541e}} , then we have
{{formula:49822ef3-9190-4f99-8372-f9295da6004d}}
Let {{formula:394cd69b-9d1e-44c7-bd81-1ca6bf64a5c9}} denote the entry at {{formula:f8fd1551-f586-4dba-b076-c349f8693392}} -th row and {{formula:ccba2107-6932-438d-9a70-e7cd7e5c8c2b}} -th column in matrix {{formula:a4eb088f-7811-4044-b497-730c415534b6}} . Let {{formula:a33661c9-1edd-4564-9619-6d14a29fc557}} denote the vector in {{formula:a16db52f-44f5-499c-861a-99d202576f07}} -th column of {{formula:10169839-0645-461c-a532-3f17b242718c}} .
We first show expectation,
{{formula:c42d6c6e-012b-4664-9249-2b32da799944}}
Secondly, we prove the variance is small
{{formula:66f1d393-b042-43cd-9eef-586844e6104d}}
where the last step follows from defining those terms to be {{formula:1fadb7bd-70f5-4945-ad5c-e527b9ea6aa6}} and {{formula:95e45a29-f2d3-47f3-8027-3ebdfd0f90cf}} . For the term {{formula:f0a41a1d-ac06-4e77-b594-97634b72e0f9}} , we have
{{formula:d25ea07b-48c6-40e3-bf9f-e6f519da951c}}
For the second term {{formula:c3b899d4-80b7-4dfe-b915-70a19446a14e}} ,
{{formula:1943c4ce-2aa4-4d88-847d-9284cad1295d}}
For the third term {{formula:c602fc96-34e8-431f-9bff-24f17f82636d}} ,
{{formula:b91cebe0-aea1-4e48-9453-91416228406c}}
Therefore, we have
{{formula:a4d72875-b630-43f0-9d14-3d1db5314b9a}}
Third, we prove the worst case bound with high probability. We can write {{formula:d6048621-22a3-4add-b2b2-b6cabaed7e30}} as follows
{{formula:ba3e69a7-ef60-4d81-8c57-c01272d79d38}}
First, we apply Khintchine's inequality, we have
{{formula:ea853b6b-cde0-4f7e-92ed-1c8744cd1d15}}
and choose {{formula:63e452ed-da1c-40b0-b691-3d0fed275a08}} .
For each {{formula:a2e142d1-35b8-4370-b9c9-7628594b51f1}} , using {{cite:25988d79d9ba449c1655ddf90a05f703b034cbbe}} we have
{{formula:a95053b7-5c04-403f-b413-3c3a51688b42}}
Taking a union bound over all {{formula:7e937a31-eab1-48ee-8e7b-d95c3a7ad148}} , we have
{{formula:98c2e751-0b01-46fd-ab69-5adde77f63a4}}
with probability {{formula:0359321d-a8e4-45b4-9e66-3f430d44a528}} .
Acknowledgments
The authors would like to thank Haotian Jiang, Swati Padmanabhan, Ruoqi Shen, Zhengyu Wang, Xin Yang and Peilin Zhong for useful discussions.
| m | 6dc9397ad25227fed6dbbfc4f0433581 |
Use of thermal infrared (TIR) cameras have always been a popular research topic in ADAS to improve night-time driving by enhancing driver’s perception. Also, infrared cameras have the advantage that they do not sense the illumination conditions, but the thermal radiations instead. Their robustness to illumination changes and shadow effects makes them useful in ADAS which is the key motivation of this research to use thermal infrared cameras for improving vehicle detection performance under night time conditions. One of the existing popular ways to use infrared images for vehicle detection purpose is to convert them into day-time RGB images, where vehicle detection already performs significantly well. This falls under image-translation task, which involves converting images from one domain (e.g., TIR image) to another (day-time RGB image). With the advancement in deep learning algorithms, a particular class of generative models, namely Generative Adversarial Networks (GANs), have become popular in generating images from random noises and also conditoned data, such as input images. The conditonal GANs (CGANs) {{cite:ace4170fb3f8dceaed233a122683dc6a4b03ddbe}} have been extensively used in image domain translation tasks {{cite:034cdfcef42399d6216e0ce94bda2ae0d7a765e8}} . These models have been successful to synthesize fake images which are realistic and similar to input images domain {{cite:ace4170fb3f8dceaed233a122683dc6a4b03ddbe}}. These conditional GANs are capable of image translation from one domain to the other given aligned data for supervised learning approaches (e.g., pix2pix) {{cite:034cdfcef42399d6216e0ce94bda2ae0d7a765e8}} and unaligned data for unsupervised learning based approaches (e.g., cycleGAN) {{cite:8fe558f85225241b4db3d303223f73ecac24d03f}} in which generated results are conditioned on input images. Such CGANs have been used in variety of tasks like semantic segmentation, grayscale visible image colourization, sketch to scene etc. Therefore, in this research we have used conditional GANs to convert TIR images to day-time RGB to leverage robustness of infrared cameras to illumination conditions.
| i | c8d3cd45e8679bcc7a755999b3706d9f |
In our experiment of the Poisson equation, where we can directly compare to traditional PINNs, we observe that is possible to obtain similar levels of errors, showing that our method can also be used for simple domains. Furthermore, we did not see a decrease in performance when we compared the numerically obtained eigenfunctions versus the exact ones. The number of eigenfunctions used did alter the accuracy of our method. This quantity is a common hyper-parameter that needs to be tuned in some sort of shape for all positional encodings {{cite:6de378f684046693f40d25892d5b7cffa1213578}}. We also observed that the size of the discretization to compute the operators also has an impact on the accuracy in a non-monotonic fashion. The way that changing the discretization affects the loss landscape and the training process requires further studies, which we believe are beyond the scope of this work.
| d | 83e5ad2a3dd1d8356496cf98d7bcb178 |
AAS can be applied to other tasks, for example, machine translation. BLEU{{cite:ebff67931a3cd799dbd4c5427456db3ec59192cd}} score used to evaluate machine translation models incorporates an average of n-gram precision but does not consider the synonymy.
Therefore, METEOR {{cite:15b20b6eb73ed18c921a0f39529dbc8b00aa31f3}} was proposed to overcome this problem. However, METEOR only relies on the synset of WordNet to get the synonyms. Our proposed AAS
has the advantage of both knowledge base and word embeddings, which would help better evaluate translation tasks.
| d | 1019a1a031e56792ae09e8e92aba8be7 |
SAC. For single-agent game, soft actor-critic (SAC) {{cite:00f83862adc7a427bb03a456a31adefcd61d06f6}} follows the maximum entropy reinforcement learning framework, which optimises the objective {{formula:f15c9b09-2c13-4aa2-a0e8-b0de4cdeda77}} to encourage the exploration {{formula:371ca24f-1adc-40f4-8973-834a117057af}} of the policy {{formula:cfd4d6bf-b670-404f-aaef-2f9f117d5602}} during the learning process. Specifically, the critic {{formula:ae750db4-5211-4753-9e7e-d245f7f6d375}} network is updated using the gradients {{formula:2f3c7560-a2c8-48de-99ec-bbc2a2dc9563}} , where {{formula:83425aef-6b30-4266-9919-b80639e36a52}} and {{formula:5ac14dbf-2a0f-4d6c-823f-4052fe8f30e3}} is the target {{formula:52d2bb3c-1530-44d6-a137-1278f28c1137}} network; the actor policy {{formula:777b8ba7-5453-4d76-8804-5d62f1243707}} is updated using the gradients {{formula:0654e4f0-12c0-4819-869d-68c0ab89e052}} .
| m | 6719a1d5748cbdae909e3e9405a69042 |
Limitations include common parameter estimation challenges when the number of model parameters and/or variables increases relative to the variables or quantities of interest for which data is available, as well as the lack of known nominal values for some model parameters in large pulmonary arteries. One challenge is non-uniqueness of parameter estimates due to the infeasability of guaranteeing a solution of the optimization problem that is a global minimum of the cost function across the parameter landscape. A second challenge is the local nature of sensitivity measures underlying the identifiability techniques used in this study, i.e. the final reduced model is not guaranteed to be unique. Indeed, the accuracy and robustness of the approaches presented in this study can be enhanced through both extended ex vivo and in vivo studies informed by the model presented here. Our approach for sensitivity and identifiability analysis and model reduction is rooted in prior works describing parameter subset selection techniques {{cite:f6752efbf903102e74b5143ee0b6714a67791fb8}}, {{cite:bb363c139e11cb4a1dca756341e3c81929a66c97}}, {{cite:a65b8b246af80865cb92e201a6299c9aa1006d18}}, {{cite:947ebcb10d5324d42a52fa1fb6e37ea9264a2069}} using an eigendecomposition of the matrix {{formula:5b197621-da25-4c75-acb2-3e2e5f151885}} , but similar results could likely be obtained using other methods.
While global sensitivity analysis techniques exist {{cite:cd08a5e86d6db04b5869c0cd10ccaf6d4a58d138}}, {{cite:5c19d9c6d73f4adf586f8dda6c1ffa25a9b92b5a}}, {{cite:f4553dfb597cea7b3ff476e9ed7eddd972a52328}}, most subset selection techniques are local. The method for identifiability analysis used here is based on eigenvalues but, as discussed in several previous studies, similar results can be obtained using other methods {{cite:8a1dfca539031a2a70b513877f90b6addf992bcd}}, {{cite:7c33ff37d730c91290fc28c4d4127ed107ed4a11}}, {{cite:de57903ba632387eb99caff43a59348a5adf668a}}. Overall, sensitivities or unidentifiable parameters for particular variables of quantities of interest can suggest which types of data will be most influential in an expanded data set. Where practical, examples of extensions include augmentation of ex vivo biomechanical testing to include measurement of the vessel opening angle, as well as incorporation of in vivo data measuring BP, flow and lumen area prior to sacrifice of the animal(s).
| d | 7a7616838b5373e38f3f6ffc3a8d656b |
Note that the DC conductivity characterises static low frequency fluctuations of the system. Recently,
the butterfly velocity {{cite:baf1d004e22742b4c294f4e3986db40e610ed75a}}, {{cite:ca84f8934d508ba514ffc9b69b7555ec7d7319ef}}, that shows how fast
fluctuations propagate in the plasma, has been studied in holographic anisotropic models
{{cite:1f9f11818c74e78febdbbc70c8dc38f984034eb8}}.
It has been observed that it exhibits a rich structure as a function of temperature, anisotropy and magnetic field and exceeds the conformal value in certain regimes. It would be interesting to investigate the butterfly's velocity for the model considered here {{cite:2e53577c5c0fa9a4de779a69be5a3062b188f3db}} as well as for light quarks holographic model {{cite:d4318f896f0ba6e829952c440def135215878d47}}. Interplane between DC conductivity and butterfly velocity over an anisotropic background has been considered recently in {{cite:b2c09719ae890c1af43f277da9dff84a25878797}}.
| d | 97bf69f9651537a28a19038cfd1aa601 |
Modern successes of deep reinforcement learning (RL) have shown promising results for control problems {{cite:d50e503a0a8c3be21445f8d0d9fee40400679a1f}}, {{cite:87d9e1b49f62833b353282120e1afd2907f3170d}}, {{cite:6915c0b59ca8ed3371b5fc2e7b22a6f98ee5f14f}}. However, training an agent for each task individually requires a large amount of task-specific environment interactions, incurring huge redundancy and prolonged human supervision. Developing algorithms that can efficiently adapt and generalize to new tasks has hence become an active area of research in the RL community.
| i | 19a6f11bac8985e2a2fab525c7f768c6 |
The aim of this paper is not to propose a new architecture with high image segmentation performance, but to investigate how different augmentation techniques affect the network's learning. Our results show that data augmentation significantly improves the performance on the validation data in many cases compared to only using patch extraction as baseline technique (see Table REF ). A possible explanation why data augmentation has not been fully explored for brain tumor segmentation is that the BraTS training set is rather large (369 subjects for the 2020 version), and several papers suggest that data augmentation would not help much {{cite:0789c5aa70c49fc2b42d900340848fb1a8aef593}}, {{cite:1b70ea9258b0dc78e8c3820c6d8dfc50a5e88dde}}, {{cite:ae66d7b2ac351dbbc9949bc9ffcd3fe8b1dd9c5f}}. Since all subjects in BraTS have been registered to a common space, augmentation is important to show the network brains from different angles, while augmentation may not be as important for a dataset where the brains have not been registered.
| d | 97bfeb7c57904a409eef0447be01fb45 |
The second step consists in looking for the extremal normalized sums of uncertainties with respect
to the involved parameters {{formula:6987412e-f788-47a4-8356-fc9d31627710}} . We start with the necessary conditions
of separability (REF ). Although not exploited analytically for {{formula:508add4d-4e68-4c17-933b-832073367c50}} , they exhibit the PPT property,
as expected from Peres' general theorem {{cite:3f5335bb74722f1df0c51f4fe3cccf13b40eecd7}}. Nevertheless, in the case of two-mode states
{{formula:58355099-e762-4482-9722-67832677114f}} , we readily recover the minimum {{formula:8c1ed3aa-dcbc-463d-a3ef-e597d7f5a782}} for {{formula:57d83e48-3478-4b0d-b939-e9cdbe20f40e}} {{cite:0a53c6f023f5d27a5663cf70c7155d789913edf1}},
as specified in Eqs. (REF ) and (REF ).
| d | 8bdd9659db6dfd5b4186a1f114f7654b |
ENet is the following regularized least squares model {{cite:2e20db6405ca9e6b798a9bf2353d168be1ae3b9e}}:
{{formula:9c42482a-dbfd-4274-85eb-7c049ab65288}}
| m | 0313ebce7501438e22c458ca3801dc2e |
A basic task is the transportation of quantum states from one location to another. Ideally one wishes to have perfect state transfer (PST) whereby a quantum state at an initial position is found with probability 1 at the final destination. While there has been seminal work on optical Bloch oscillations in waveguide arrays, see for instance {{cite:e675b9ca612906d7534bc6810488caa8c47cb984}}, {{cite:ea60f3f35ee70cfe78d683a2602074e267ef8194}}, {{cite:5a77150d0e7084570da42ddba968870c3cf2e97f}}, this review focuses on coherent transport in spin-inspired photonic lattices. Since it has been suggested that spin chains {{cite:7632d7e15366e887c7a3d6d16542fd1dd498db87}} with properly modulated couplings {{cite:eb3db8c556ca92b09370add2cd7510fbd628be03}}, {{cite:3416aa98e91eb0c69119d3d5266957808f98b452}} could move a qubit from end-to-end with probability 1 in a given time, there has been enormous interest in this question and indeed various analytic models have been found {{cite:9d4708db3bede19748d9eb9128dbbae6c89a2e76}}, {{cite:4139a03cd14f516a6299b69aa780329e83d1f1e0}}, {{cite:c93f0a1a3bc38d10aac081797be69ac4fda52b27}}, {{cite:bbde6f482cc5d39d90832e1aca602cd95e4ce7ea}}, {{cite:51298065d24ecf91abdfb82ffa2ea5f9c52145a5}}, {{cite:175e9d9647588cb74dfa154232641307923bd1c1}}, {{cite:689c876ac37cdf91f1a094167c5afee3ed020e02}}, {{cite:fae4a568420f421d682ca20d7029948d225553a6}}. See {{cite:e606a0ef55063def3ab35cb2a2def93a2f2c550d}}, {{cite:fee224462db0da070e9a05ccb66776f5969fe9ca}}, {{cite:aae188253ba3520d998e4bdc73cb89df8bf8b38c}} for reviews. Remarkably experimental verifications have been offered {{cite:58c362921001154ac4afaa0926b1f1ec581b0f56}}, {{cite:5bf60ada6ac0a242729298a093a8e386d9ac6ab3}}, {{cite:2bc608502d084974d3b01aefea3ea4211c2e868a}} for the simplest (Krawtchouk) model with PST by using arrays of evanescently coupled waveguides {{cite:a37302071cc713e41baef4ec5994811aff610a76}}, {{cite:b3e7e4a16a69db9b10a74d0a210e0fbce4e48bbe}}, {{cite:53b80c42c34a8c5a411f9811b796e3aed1b59c94}}. These can be pictured as a set of optical fibers stacked side by side in a plane. The point is that in the nearest-neighbour approximation, the equations of coupled mode theory describing the propagation of a single photon in waveguide arrays are mathematically identical to those governing the one-excitation dynamics of a certain spin chain with non-uniform couplings. Each qubit is represented by a single waveguide and the presence or absence of a photon in a waveguide corresponds to the {{formula:70e590d3-c32f-4b10-bd0b-30034e1ae1a5}} or {{formula:a63b0d3c-04d3-4e87-9fb1-4063c0f88f5d}} state of the qubit at the corresponding site. The evolution time {{formula:ca054d2c-0a28-45cc-adaf-ac47c8ebac82}} of the spin chain is identified with the propagation distance {{formula:26e6de60-81d0-464c-895b-dbdbc2cc805a}} along the waveguides. Therefore PST is tantamount to finding in the last waveguide after a distance {{formula:2b3fe703-2fe1-4372-a74e-62c008144246}} , the photon initially inserted (at {{formula:c884a26b-ccf1-444f-b0cc-6a1820dc680c}} ) in the first fiber of the arrayStrictly speaking in the photonic lattice realizations, one is producing an excitation transfer. In a spin chain, a state, i.e., an arbitrary combination of spin up and down will be transported. This is modeled in {{cite:5bf60ada6ac0a242729298a093a8e386d9ac6ab3}} in an optical settings by encoding a state in polarizations.. The required couplings between the waveguides are engineered by adjusting the distances so that they match the theoretical values. The analysis that underscores the design of the spin chains/photonic lattices with PST shows that the couplings and local magnetic fields/propagation constants correspond to the recurrence coefficients of orthogonal polynomials that are naturally associated to these systems. The analytic models in fact correspond to orthogonal polynomials that have been fully characterized {{cite:0c8a2caccad33083ffb50499cc1fe84fe4552228}} and they are thus referred to by the name of the polynomials that are attached to them. We have already hinted at that above when we mentioned the Krawtchouk model.
| i | 3a2c992da059aefa99a4bf4c57a686d7 |
The most accurate current denoisers are trained on a representative dataset consisting of noisy images and their clean counterparts. The earliest method to do this with a neural network was {{cite:25b9290bc21eb020a3be3f8a5d4a8e46b2e04c7c}}. This was heavily refined in both the works of Mao et al. {{cite:e8fc6e4bf348a0f96a25d39338f0fe6472e22c2d}} and Zhang et al. {{cite:faec358c564204687bfd276973007e344a2bbada}} to achieve results that are still among the best achievable today. A more recent approach is {{cite:20a0a32f2452bd23148aeede193dd8e4bef579db}} which uses a two-module approach to push accuracy even further.
| m | f87de6c1c3f0f90546de64c9b0085e53 |
A comparison between the abundances of ethanol and “warm" glycolaldehyde relative to that of {{formula:fa1a0cb4-601c-4a8b-8c29-1e53d674fb5c}} for each position is shown in Figure REF .
We found that {{formula:1c004167-1140-4428-891c-dba54c56b10e}} , with a correlation coefficient of 0.96. This result agrees well with the relation found for star-forming regions in the Galactic disk: L1157-b1 {{cite:ff233853d3f919eb533ae4cd149e19bf2f6adbdb}}, IRAS 16293-2422 {{cite:2786481f72ee24b52671e428f10bc95b73fe520a}}, IRAS 2A and IRAS 4A {{cite:27bf9362479c9f4e8a8c0daf832afcb08c97531b}} (see Figure REF ). The strong correlation between ethanol and “warm" glycolaldehyde suggests that the “warm" glycolaldehyde may be chemically related to ethanol. Recently, theoretical studies have shown that a new gas-phase scheme of reactions, involving ethanol as a parent molecule, can lead to the formation of glycolaldehyde and formic acid {{cite:080f5714acb1c15b2534814ce8189d48e519951b}}. This model can well explain the abundance correlation between “warm" glycolaldehyde and ethanol. This model is further supported by the strong correlation between the abundances of ethanol and formic acid relative to that of {{formula:d22cf1c7-9d6c-48a8-ae7a-a219f3ad05f4}} (Figure REF ), with a correlation coefficient of 0.92. The correlation between the abundances relative to H{{formula:a0f1cc1e-d5f8-4fc7-9ab2-3b983fbf1d09}} of ethanol, “warm" glycolaldehyde and formic acid suggests that “warm" glycolaldehyde and formic acid are chemically related to ethanol and are possibly produced by ethanol via gas-phase reactions {{cite:080f5714acb1c15b2534814ce8189d48e519951b}}. However, we couldnot exclude the possibility that these molecules are co-spatial in the same gas and trace a warm gas phase. Interferometric observations of a large sample of interstellar sources could help to investigate whether ethanol, formic acid and “warm" glycolaldehyde are co-spatial and constrain their formation pathways {{cite:ec53dc9fb21909b723839b15fd00265f88b32fb4}}.
{{figure:461cdf27-c612-4e27-bee7-fa5c73ad5749}} | d | 53f3adf967d06924fd4fc9ce5e01475b |
In recent years, deep learning based approaches have burgeoned in the field of medical image registration {{cite:22488ff81589bc6f6f5921bb5a135031c8df4fe1}}. Their success has been largely driven by their exceptionally fast inference speeds. The most effective methods, such as VoxelMorph {{cite:f2b395a49d049c30df1ba4b506440714e45afcbf}}, usually adopt a U-Net style architecture to estimate dense, spatial deformation fields. They only require one forward propagation during inference, and thus can register images several orders of magnitudes faster than traditional iterative methods. Following the success of VoxelMorph, a large number of deep learning based approaches have been proposed for various registration tasks {{cite:5966bf2bbd490ff09411ebb210775ec26a760973}}, {{cite:4da84f7da9b8d8eefc74105b1f90e4aaec414c95}}, {{cite:eeef3cf23f7ec264a144c6c4940a0d7eff2b40e0}}, {{cite:99e9ae9b5baa071a634fa3c2106056f1ee80b516}}, {{cite:f8b73a2183998fe4fd0983aa24b91a45282ab863}}, {{cite:3f396aefee3efd925e64409ea15e4438d3f98990}}, {{cite:f1f6a41c7f55067d8bae40deea546467dc23feab}}. These models either use multiple U-Net style networks in a cascaded way or replace basic convolution blocks in VoxelMorph with more sophisticated ones such as swin-transformers {{cite:3f396aefee3efd925e64409ea15e4438d3f98990}} to boost registration performance. However, these changes rapidly increase the number of network parameters and multiply-add operations (mult-adds), sacrificing training and inference efficiency altogether.
| i | 84a35b5d62c74369fb25dac083af6334 |
There are many stories surrounding the origin of the Chicken McNugget theorem. However, the most popular by far remains that of the Chicken McNugget. Originally, McDonald's sold its nuggets in packs of 9 and 20. Thus, to find the largest number of nuggets that could not have been bought with these packs creates the Chicken McNugget Theorem (the answer worked out to be 151 nuggets). More description on the history of McNugget problem can be found, for example, in {{cite:03cd8480b063b12ae25447d0659b6713817b2859}}, {{cite:9b7443696e96966959987e473b0e1c0738a96ec6}}.
| i | 65cb2a5bb48386c6a2a470894e5ae4f8 |
Considering how the presence of confinement affects the conformational dynamics of a chain{{cite:2ff6c4e378f75ea4decf9612eb79a017bf330796}}, {{cite:fc7b7b0a1db4d17db37cb3f25966b9e1efc31474}}, {{cite:04693fecbfdd6b3c3ed1bb090caa2d58571861a3}}, its outright indispensable to look at the influence of the geometric constraint. Figure REF -a elucidates the field induced compression of a semi-flexible polymer ({{formula:21a70b72-1913-4cbc-a50e-254f82999be6}} ), for a range of varying pore radii. A striking observation is that, while moderately confined polymer, {{formula:8b0c581b-a095-489c-bc6e-2dc20f7fd282}} , undergoes a substantial compression at lower field strengths followed by a stretching, chain under weak confinement/tending to bulk {{formula:7eb1637f-c9ec-4835-b32f-dc135525a462}} exhibits no compression and undergoes a monotonic stretching. This is evident in {{formula:77e307f2-0f8e-4c74-afcb-0b3fe636e123}} where a compressive dip in {{formula:a072c5c0-edd0-496e-a85b-bd712357e646}} is seen for lower G values, while for {{formula:0b11c962-967c-41c7-ad45-f2d9ead8e994}} only monotonic stretching is seen, devoid of any shrinkage. A heuristic scaling obtained in Fig. REF -a for the field {{formula:20813ca9-1ae7-45a1-af8e-023851a9b925}} beyond which stretching is effectuated, gives a {{formula:5daa80bb-162e-49af-b990-9e8599a6afb4}} dependence.
The deviation in {{formula:8a141025-5ef2-4d6f-a6b8-828767cfdcc7}} scaling seen for {{formula:b9b52c4a-be65-4761-aacd-4dc398d4441a}} , we speculate is a repercussion of {{formula:2dec2ff5-b73e-4a68-850e-bfd9f755f310}} hitting a different regime, where the effect of confinement diminishes such that all {{formula:f982eef5-621b-4818-af31-08f66fed5ebe}} tend to the bulk behaviour. This is elucidated in Fig.REF -b, which shows the structural variation of a chain under varying confinement in equilibrium. It shows a typical non-monotonous pattern, as reported earlier{{cite:5cfaf1654d6b6908429e1f594609f9caac6e49c5}}. The chain slightly swells back to its bulk value beyond {{formula:8800a3e5-bbcf-4ae7-98cf-9fa85bc245b0}} .
{{figure:a9f3d819-6116-4e92-82b5-e015649703d7}} | r | 97b8ef444f3cacfc80fabca3e6f2484f |
Note that for SDE, {{formula:7a612ec3-4b11-41a6-acca-ed818aa177e0}} comes from two discrete approximations with different timesteps but the same Brownian path. To generate random samples {{formula:134eabd1-a276-4b65-b414-0a3c7091dc21}} , one method suggested in {{cite:b17d77b9e238eb083d44bed2956bfc49d2a951c3}} is as follows. First constructing the Brownian increments for the simulation of the discrete path leading to the evaluation of {{formula:522a0d4b-db31-4ef2-8a8d-12254e2042ae}} . Then summing them in groups of size 2 to give the discrete Brownian increments for the evaluation of {{formula:cad28b1e-caae-44dc-ade4-538806b20df8}} .
| m | de2df3adfab7bb52d830b75f918d1b57 |
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