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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+1
+Knockoffs-SPR: Clean Sample Selection in
+Learning with Noisy Labels
+Yikai Wang, Yanwei Fu, and Xinwei Sun.
+Abstract—A noisy training set usually leads to the degradation of the generalization and robustness of neural networks. In this paper,
+we propose a novel theoretically guaranteed clean sample selection framework for learning with noisy labels. Specifically, we first
+present a Scalable Penalized Regression (SPR) method, to model the linear relation between network features and one-hot labels. In
+SPR, the clean data are identified by the zero mean-shift parameters solved in the regression model. We theoretically show that SPR
+can recover clean data under some conditions. Under general scenarios, the conditions may be no longer satisfied; and some noisy
+data are falsely selected as clean data. To solve this problem, we propose a data-adaptive method for Scalable Penalized Regression
+with Knockoff filters (Knockoffs-SPR), which is provable to control the False-Selection-Rate (FSR) in the selected clean data. To
+improve the efficiency, we further present a split algorithm that divides the whole training set into small pieces that can be solved in
+parallel to make the framework scalable to large datasets. While Knockoffs-SPR can be regarded as a sample selection module for a
+standard supervised training pipeline, we further combine it with a semi-supervised algorithm to exploit the support of noisy data as
+unlabeled data. Experimental results on several benchmark datasets and real-world noisy datasets show the effectiveness of our
+framework and validate the theoretical results of Knockoffs-SPR. Our code and pre-trained models will be released.
+Index Terms—Learning with Noisy Labels, Knockoffs Method, Type-Two Error Control.
+!
+1
+INTRODUCTION
+D
+EEP learning has achieved remarkable success on
+many supervised learning tasks trained by millions
+of labeled training data. The performance of deep models
+heavily relies on the quality of label annotation since neural
+networks are susceptible to noisy labels and even can easily
+memorize randomly labeled annotations [1]. Such noisy
+labels can lead to the degradation of the generalization
+and robustness of such models. Critically, it is expensive
+and difficult to obtain precise labels in many real-world
+scenarios, thus exposing a realistic challenge for supervised
+deep models to learn with noisy data.
+There are many previous efforts in tackling this challenge by
+making the models robust to noisy data, such as modifying
+the network architectures [2]–[5] or loss functions [6]–[9].
+This paper addresses the challenge by directly selecting
+clean samples. Inspired by the dynamic sample selection
+methods [9]–[16], we construct a “virtuous” cycle between
+sample selection and network training: the selected clean
+samples can improve the network training; and on the
+other hand, the improved network has a more powerful
+ability in picking up clean data. As this cycle evolves, the
+performance can be improved. To well establish this cycle, a
+key question remains: how to effectively differentiate clean data
+from noisy ones?
+Preliminary. Typical principles in existing works [9]–[16]
+to differentiate clean data from noisy data include large
+loss [11], inconsistent prediction [17], and irregular feature
+•
+Yikai Wang and Yanwei Fu contribute equally.
+•
+Xinwei Sun is the corresponding author.
+•
+Yikai Wang, Yanwei Fu and Xinwei Sun are with the School of
+Data Science, Fudan University. E-mail: {yikaiwang19, yanweifu,
+sunxinwei}@fudan.edu.cn
+representation [18]. The former two principles identify
+irregular behaviors in the label space, while the last one
+analyzes the instance representations of the same class in
+the feature space. In this paper, we propose unifying the
+label and feature space by making the linear relationship,
+yi = x⊤
+i β + ε,
+(1)
+between feature-label pair (xi ∈ Rp: feature vector; yi ∈ Rc:
+one-hot label vector) of data i. We also have the fixed
+(unknown) coefficient matrix β ∈ Rp×c, and random noise
+ε ∈ Rc. Essentially, the linear relationship here is an ideal
+approximation, as the networks are trained to minimize
+the divergence between a (soft-max) linear projection of
+the feature and a one-hot label vector. For a well-trained
+network, the output prediction of clean data is expected
+to be as similar to a one-hot vector as possible, while the
+entropy of the output of noisy data should be large. Thus if
+the underlying linear relation is well-approximated without
+soft-max operation, the corresponding data is likely to be
+clean. In contrast, the feature-label pair of noisy data may
+not be approximated well by the linear model.
+The simplest way to measure the goodness of the linear
+model in fitting the feature-label pair is to check the
+prediction error, or residual, ri = yi − x⊤
+i ˆβ, where ˆβ is the
+estimate of β. The larger ∥r∥ indicates a larger fitting error
+and thus more possibility for instance i to be outlier/noisy
+data. Many methods have been proposed to test whether ri
+is non-zero. Particularly, we highlight the classical statistical
+leave-one-out approach [19] that computes the studentized
+residual as,
+ti =
+yi − x⊤
+i ˆβ−i
+ˆσ−i
+�
+1 + x⊤
+i
+�X⊤
+−iX−i
+�−1 xi
+�1/2 ,
+(2)
+arXiv:2301.00545v1  [cs.LG]  2 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+2
+Images
+Features
+Noisy Labels
+Stage 1:
+Network Learning
+Classifier
+Stage 2:
+Sample Selection
+β ∈ Rd×c
+Noisy Labels
+Y ∈ Rn×c
+X ∈ Rn×d
+Features
+Noisy Data Indicator
+=
++
+γ ∈ Rn×c
+Yi (1, 0, 0)
+(0, 1, 0) ˜Yi
+Permute
+Compare
+Selected Clean Data
+Fig. 1. Knockoffs-SPR runs a cycle between network learning and sample selection, where clean data are selected via the comparison of the
+mean-shift parameters between its original label and permuted label.
+where ˆσ is the scale estimate and the subscript −i indicates
+estimates based on the n − 1 observations, leaving out
+the i-th data for testing. Equivalently, the linear regression
+model can be re-formulated into explicitly representing the
+residual,
+Y = Xβ + γ + ε,
+εi,j ∼ N(0, σ2),
+(3)
+by introducing a mean-shift parameter γ as in [20] with the
+feature X ∈ Rn×p, and label Y ∈ Rn×c paired and stacked
+by rows. For each row of γ ∈ Rn×c, γi represents the predict
+residual of the corresponding data. This formulation has
+been widely studied in different research topics, including
+economics [21]–[24], robust regression [20], [25], statistical
+ranking [26], face recognition [27], semi-supervised few-shot
+learning [28], [29], and Bayesian preference learning [30],
+to name a few. This formulation is differently focused on
+the specific research tasks. For example, for the robust
+regression problem [20], [25], the target is to get a robust
+estimate ˆβ against the influence of γ. Here for solving the
+problem of learning with noisy labels, we are interested
+in recovering zeros elements of γ, since these elements
+correspond to clean data.
+SPR [31]. To this end, from the statistical perspective,
+our conference report [31] starts from Eq. (3) to build up
+a sample selection framework, dubbed Scalable Penalized
+Regression (SPR). With a sparse penalty P(γ; λ) on γ, the
+SPR obtains a regularization solution path of γ(λ) by
+evolving λ from ∞ to 0. Then it identifies those samples
+that are earlier (or at larger λ) selected to be non-zeros
+as noisy data and those later selected as clean data, with
+a manually specified ratio of selected data. Under the
+irrepresentable condition [4], [33], the SPR enjoys model
+selection consistency in the sense that it can recover the set
+of noisy data. By feeding only clean data into next-round
+training, the trained network is less corrupted by the noisy
+data and hence performs well empirically.
+Knockoffs-SPR. However, the irrepresentable condition
+demands the prior of the ground-truth noisy set, which
+is not accessible in practice. When this condition fails,
+the trained network with SPR may be still corrupted by
+a large proportion of noisy data, leading to performance
+degradation as empirically verified in our experiments. To
+amend this problem, we provide a data-adaptive sample
+selection algorithm, in order to well control the expected
+rate of noisy data in the selected data under the desired level
+q, e.g., q = 0.05. As the goal is to identify clean data for the
+next-round training, we term this rate as the False-Selection-
+Rate (FSR). The FSR is the expected rate of the type-II error
+in sparse regression, as non-zero elements correspond to
+the noisy data. Our method to achieve the FSR control is
+inspired by the ideas of Knockoffs in Statistics, which is
+a recently developed framework for variable selection [1],
+[2], [34], [35]. The Knockoffs framework aims at selecting
+non-null variables and controlling the False-Discovery-Rate
+(FDR), by taking as negative controls knockoff features ˜
+X,
+which are constructed as a fake copy for the original features
+X. Here, the FDR corresponds to the expectation of the
+type-I error rate in sparse regression. Therefore, the vanilla
+Knockoffs cannot be directly applied to our SPR framework,
+since FSR is the expected rate of the type-II error and there
+is no theoretical guarantee in Knockoffs for this control. To
+achieve the FSR control, we propose Knockoffs-SPR, which
+turns to construct the knockoff labels ˜Y via permutation for
+the original label Y , and incorporates it into a data-partition
+strategy for FSR control.
+Formally, we repurpose the knockoffs in Statistics in our
+SPR method; and propose a novel data-adaptive sample
+selection algorithm, dubbed Knockoffs-SPR. It extends SPR
+in controlling the ratio of noisy data among the selected
+clean data. With this new property, Knockoffs-SPR ensures
+that the clean pattern is dominant in the data and hence
+leads to better network training. Specifically, we partition
+the whole noisy training set into two random subsets and
+apply the Knockoffs-SPR to two subsets separately. For each
+time, we use one subset to estimate the intercept β and the
+other to select the clean data by comparing between the
+solution paths of γ(λ) and ˜γ(λ) that respectively obtained
+via regression on noisy labels and the permuted labels. With
+such a decoupled structure between β and γ, we prove
+that the FSR can be controlled by any prescribed level.
+Compared with the original theory of SPR, our new theory
+enables us to effectively select clean data under general
+conditions. Besides, Knockoffs-SPR also enjoys a superior
+performance over the original SPR.
+Together with network training, the whole framework is
+illustrated in Fig. 1 in which the sample selection and
+the network learning are well incorporated into each
+other. Specifically, we run the network learning process
+and sample selection process iteratively and repeat this
+cycle until convergence. To incorporate Knockoffs-SPR into
+the end-to-end training pipeline of deep architecture, the
+simplest way is to directly solve Knockoffs-SPR for each
+
+featureA
+B
+C
+DJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+3
+training mini-batch or training epoch to select clean data.
+Solving Knockoffs-SPR for each mini-batch is efficient but
+suffers from the identifiability issue. The sample size in a
+mini-batch may be too small to distinguish clean patterns
+from noisy ones among all classes, especially for large
+datasets with small batch size. Solving Knockoffs-SPR for
+the whole training set is powerful but suffers from the
+complexity issue, leading to an unacceptable computation
+cost. To resolve these two problems, we strike a balance
+between complexity and identifiability by proposing a
+splitting strategy that divides the whole data into small
+pieces such that each piece is class-balanced with the proper
+sample size. In this regard, the sample size of each piece is
+small enough to be solved efficiently and large enough to
+distinguish clean patterns from noisy ones. Then Knockoffs-
+SPR runs on each piece in parallel, making it scalable to
+large datasets.
+As the removed noisy data still contain useful information
+for network training, we adopt the semi-supervised training
+pipeline with CutMix [38] where the noisy data are utilized
+as unlabeled data. We conduct extensive experiments to
+validate the effectiveness of our framework on several
+benchmark datasets and real-world noisy datasets. The
+results show the efficacy of our Knockoffs-SPR algorithm.
+Contributions. Our contributions are as follows:
+• Ideologically, we propose to control the False-Selection-
+Rate in selecting clean data, under general scenarios.
+• Methodologically, we propose Knockoffs-SPR, a data-
+adaptive method to control the FSR.
+• Theoretically, we prove that the Knockoffs-SPR can
+control the FSR under any desired level.
+• Algorithmically, we propose a splitting algorithm for
+better sample selection with balanced identifiability and
+complexity in large datasets.
+• Experimentally, we demonstrate the effectiveness and
+efficiency of our method on several benchmark datasets
+and real-world noisy datasets.
+Extensions. Our conference version of this work, SPR, was
+published in [31]. Compared with SPR [31], we have the
+following extensions.
+• We identify the limitation of the SPR and consider the FSR
+control in selecting clean data.
+• We propose a new framework: Knockoffs-SPR which is
+effective in selecting clean data under general scenarios,
+theoretically and empirically.
+• We apply our method on Clothing1M and achieve better
+results than compared baselines.
+Logistics. The rest of this paper is organized as follows:
+• In Section 9, we introduce our SPR algorithm with its
+noisy set recovery theory.
+• In Section 3, the Knockoffs-SPR algorithm is introduced
+with its FSR control theorem.
+• In Section 4, several training strategies are proposed to
+well incorporate the Knockoffs-SPR with the network
+training.
+• In Section 5, connections are made between our proposed
+works and several previous works.
+• In Section 6, we conduct experiments on several synthetic
+and real-world noisy datasets.
+• Section 7 concludes this paper.
+2
+CLEAN SAMPLE SELECTION
+2.1
+Problem Setup
+We are given a dataset of image-label pairs {(imgi, yi)}n
+i=1,
+where the noisy label yi is corrupted from the ground-
+truth label y∗
+i . The ground-truth label y∗
+i and the corruption
+process are unknown. Our target is to learn a recognition
+model f(·) such that it can recognize the true category y∗
+i
+from the image imgi, i.e., f(imgi) = y∗
+i , after training on the
+noisy label yi.
+In this paper, we adopt deep neural networks as the
+recognition model and divide the f(·) into fc(g(·)) where
+g(·) is the deep model for feature extraction and fc(·) is
+the final fully-connected layer for classification. For each
+input image imgi, the feature extractor g(·) is used to encode
+the feature xi := g(imgi). Then the fully-connected layer is
+used to output the score vector ˆyi = fc(xi) which indicates
+the chance it belongs to each class and the prediction is
+provided with ˆyi = argmax(ˆyi).
+As the training data contain many noisy labels, simply
+training from all the data leads to severe degradation
+of generalization and robustness. Intuitively, if we could
+identify the clean labels from the noisy training set, and
+train the network with the clean data, we can reduce the
+influence of noisy labels and achieve better performance and
+robustness of the model. To achieve this, we thus propose a
+sample selection algorithm to identify the clean data in the
+noisy training set with theoretical guarantees.
+Notation. In this paper, we will use a to represent scalar, a
+to represent a vector, and A to represent a matrix. We will
+annotate a∗ to denote the ground-truth value of a. We use
+∥ · ∥F to denote the Frobenius norm.
+2.2
+Clean Sample Selection via Penalized Regression
+Motivated
+by
+the
+leave-one-out
+approach
+for
+outlier
+detection, we introduce an explicit noisy data indicator γi
+for each data and assume a linear relation between extracted
+feature xi and one-hot label yi with noisy data indicator as,
+yi = x⊤
+i β + γi + εi,
+(4)
+where yi ∈ Rc is one-hot vector; and xi ∈ Rp, β ∈
+Rp×c, γi ∈ Rc, εi ∈ Rc. The noisy data indicator γi can be
+regarded as the correction of the linear prediction. For clean
+data, yi ∼ N(x⊤
+i β∗, σ2Ic) with γ∗
+i = 0, and for noisy data
+y∗
+i = yi −γ∗
+i ∼ N(x⊤
+i β∗, σ2). We denote C := {i : γ∗
+i = 0}
+as the ground-truth clean set.
+To select clean data for training, we propose Scalable
+Penalized Regression (SPR), designed as the following sparse
+learning paradigm,
+argmin
+β,γ
+1
+2 ∥Y − Xβ − γ∥2
+F + P(γ; λ),
+(5)
+where we have the matrix formulation X ∈ Rn×p, and
+Y
+∈
+Rn×c of {xi, yi}n
+i=1; and P(·; λ) is a row-wise
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+4
+Fig. 2. Solution Path of SPR. Red lines indicate noisy data while blue
+lines indicate clean data. As λ decreases, the γi gradually solved with
+non-zero values.
+sparse penalty with coefficient parameter λ. So we have
+P(γ; λ) = �n
+j=1 P(γi; λ), e.g., group-lasso sparsity with
+P(γ; λ) = λ �
+i ∥γi∥2.
+To estimate C, we only need to solve γ with no need to
+estimate β. Thus to simplify the optimization, we substitute
+the Ordinary Least Squares (OLS) estimate for β with γ
+fixed into Eq. (5). To ensure that ˆβ is identifiable, we apply
+PCA on X to make p ≪ n so that the X has full-column
+rank. Denote
+˜
+X = I − X
+�X⊤X
+�† X⊤, ˜Y
+=
+˜
+XY , the
+Eq. (5) is transformed into
+argmin
+γ
+1
+2
+��� ˜Y − ˜
+Xγ
+���
+2
+F + P(γ; λ),
+(6)
+which is a standard sparse linear regression for γ. Note that
+in practice we can hardly choose a proper λ that works well
+in all scenarios. Furthermore, from the equivalence between
+the penalized regression problem and Huber’s M-estimate,
+the solution of γ is returned with soft-thresholding. Thus
+it is not worth finding the precise solution of a single γ.
+Instead, we use a block-wise descent algorithm [39] to solve
+γ with a list of λs and generate the solution path. As
+λ changes from ∞ to 0, the influence of sparse penalty
+decreases, and γi are gradually solved with non-zero values,
+in other words, selected by the model, as visualized in
+Fig. 2. Since earlier selected instance is more possible to be
+noisy, we rank all samples in the descendent order of their
+selecting time defined as:
+Zi = sup {λ : γi (λ) ̸= 0} .
+(7)
+A large Zi means that the γi is earlier selected. Then the top
+samples are identified as noisy data and the other samples
+are selected as clean data. In practice, we select 50% of the
+data as clean data.
+2.3
+The Theory of Noisy Set Recovery in SPR
+The SPR enjoys theoretical guarantees that the noisy data
+set can be fully recovered with high probability, under
+the irrepresentable condition [33]. Formally, consider the
+vectorized version of Eq. (6):
+argmin
+⃗γ
+1
+2
+���⃗y − ˚
+X⃗γ
+���
+2
+2 + λ ∥⃗γ∥1 ,
+(8)
+where ⃗y, ⃗γ is vectorized from Y , γ in Eq. (6); ˚
+X = Ic ⊗ ˜
+X
+with ⊗ denoting the Kronecker product operator. Denote
+S := supp(⃗γ∗), which is the noisy set Cc. We further denote
+˚
+XS (resp. ˚
+XSc) as the column vectors of ˚
+X whose indexes
+are in S (resp. Sc) and µ ˚
+X = maxi∈Sc ∥ ˚
+X∥2
+2. Then we have
+Theorem 1 (Noisy set recovery). Assume that:
+C1, Restricted eigenvalue: λmin( ˚
+X⊤
+S ˚
+XS) = Cmin > 0;
+C2, Irrepresentability: there exists a η ∈ (0, 1], such that
+∥ ˚
+X⊤
+Sc ˚
+XS( ˚
+X⊤
+S ˚
+XS)−1∥∞ ≤ 1 − η;
+C3, Large error:
+⃗γ∗
+min := mini∈S |⃗γ∗
+i | > h(λ, η, ˚
+X, ⃗γ∗);
+where ∥A∥∞
+:=
+maxi
+�
+j |Ai,j|, and h(λ, η, ˚
+X, ⃗γ∗)
+=
+λη/�Cminµ ˚
+X + λ∥( ˚
+X⊤
+S ˚
+XS)−1sign(⃗γ∗
+S)∥∞.
+Let λ ≥
+2σ√µ ˚
+X
+η
+√log cn. Then with probability greater than
+1 − 2(cn)−1, model Eq. (8) has a unique solution ˆ⃗γ such that: 1)
+If C1 and C2 hold, ˆ
+Cc ⊆ Cc;2) If C1, C2 and C3 hold, ˆ
+Cc = Cc.
+We present the proof in the appendix, following the
+treatment in [4], [40]. In this theorem, C1 is necessary to get
+a unique solution, and in our case is mostly satisfied with
+the natural assumption that the clean data is the majority
+in the training data. If C2 holds, the estimated noisy data
+is the subset of truly noisy data. This condition is the key
+to ensuring the success of SPR, which requires divergence
+between clean and noisy data such that we cannot represent
+clean data with noisy data. If C3 further holds, the estimated
+noisy data is exactly all the truly noisy data. C3 requires the
+error measured by γi is large enough to be identified from
+random noise. If the conditions fail, SPR will fail in a non-
+vanishing probability, not deterministic.
+3
+CONTROLLED CLEAN SAMPLE SELECTION
+In the last section, we stop the solution path at λ such
+that 50% samples are selected as clean data. If this happens
+to be the rate of clean data, Thm. 1 shows that our SPR
+can identify the clean data C under the irrepresentable
+condition. However, the irrepresentable condition and the
+information of the ground-truth clean set C are practically
+unknown, making this theory hard to be used in the real
+life. Particularly, with |Cc| unknown, the algorithm can stop
+at an improper time such that the noisy rate of the selected
+clean data ˆC can be still high, making the next-round trained
+model corrupted a lot by noisy patterns.
+To resolve the problem of false selection in SPR , we in this
+section propose a data-adaptive early stopping method for
+the solution path, that targets controlling the expected noisy
+rate of the selected data dubbed as False-Selection-Rate (FSR)
+under the desired level q (0 < q < 1):
+FSR = E
+�
+�#
+�
+j : j ̸∈ H0 ∩ ˆC
+�
+#
+�
+j : j ∈ ˆC
+�
+∨ 1
+�
+� ,
+(9)
+where ˆC = {j : ˆγj = 0} is the recovered clean set of
+γ, and H0 : γ∗
+i = 0 denotes the null hypothesis, i.e., the
+sample i belonging to the clean dataset. Therefore, the FSR
+in Eq. (9) targets controlling the false rate among selected
+null hypotheses, which is also called the expected rate of
+the type-II error in hypothesis testing.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+5
+3.1
+Knockoffs-SPR
+To achieve the FSR control, we propose the Knockoffs-
+SPR for clean sample selection. Our method is inspired
+by knockoff methods [1], [2], [34], [35], [41] with the
+different focus that we target selecting clean labels via
+permutation instead of constructing knockoff features to
+select explanatory variables. Specifically, under model (4)
+we permute the label for each data and construct the
+permutation ˜y. Then model (4) can be solved for y and
+˜y to obtain the solution paths γ(λ) and ˜γ(λ), respectively.
+We will show that this construction can pick up clean data
+from noisy ones, by comparing the selecting time (Eq. (7))
+between γ(λ) and ˜γ(λ) for each data. On the basis of this
+construction, we propose to partition the whole dataset
+into two disjoint parts, with one part for estimating β
+and the other for learning γ(λ) and ˜γ(λ). We will show
+that the independent structure with such a data partition
+enables us to construct the comparison statistics whose sign
+patterns among alternative hypotheses (noisy data) are the
+independent Bernoulli processes, which is crucial for FSR
+control.
+Formally speaking, we split the whole data D into D1 :=
+(X1, Y1) and D2
+:=
+(X2, Y2) with ni
+:=
+|Di|, and
+implement Knockoffs-SPR on both D1 and D2. In the
+following, we only introduce the procedure on D2, as the
+procedure for D1 shares the same spirit. Roughly speaking,
+the procedure is composed of three steps: i) estimate β on
+D1; ii) estimate ˜γ(λ)) on D2; and iii) construct the comparison
+statistics and selection filters. We leave detailed discussions for
+each step in Sec. 3.2.
+Step i): Estimating β on D1. Our target is to provide
+an estimate of β that is independent of D2. The simplest
+strategy is to use the standard OLS estimator to obtain
+ˆβ1. However, this estimator may not be accurate since it
+is corrupted by noisy samples. For this consideration, we
+first run SPR on D1 to get clean data and then solve β via
+OLS on the estimated clean data.
+Step ii): Estimating (γ(λ), ˜γ(λ)) on D2. After obtaining the
+solution ˆβ1 on D1 , we learn the γ(λ) on D2:
+1
+2
+���Y2 − X2 ˆβ1 − γ2
+���
+2
+F + P(γ2; λ).
+(10)
+For each one-hot encoded vector y2,j, we randomly permute
+the position of 1 and obtain another one-hot vector ˜y2,j ̸=
+y2,j. For clean data j, the ˜y2,j turns to be a noisy label;
+while for noisy data, the ˜y2,j is switched to another noisy
+label with probability c−2
+c−1 or clean label with probability
+1
+c−1. After obtaining the permuted matrix as ˜Y2, we learn
+the solution paths (γ2(λ), ˜γ2(λ)) using the same algorithm
+as SPR via:
+�
+�
+�
+�
+�
+1
+2
+���Y2 − X2 ˜β1 − γ2
+���
+2
+F + �
+j P(γ2,j; λ),
+1
+2
+��� ˜Y2 − X2 ˜β1 − ˜γ2
+���
+2
+F + �
+j P(˜γ2,j; λ).
+(11)
+Step iii): Comparison statistics and selection filters.
+After obtaining the solution path (γ2(λ), ˜γ2(λ)), we define
+sample significance scores with respect to y2,i and ˜y2,i of
+each i respectively, as the selection time: Zi := sup{λ :
+Algorithm 1 Knockoffs-SPR
+Input: subsets D1 and D2.
+Output: clean set of D2.
+1: Use D1 to fit an linear regression model and get β(D1);
+2: Generate permuted label of each sample i in D2;
+3: Solve Eq. (26) for D2 and generate {Wi} by Eq. (12);
+4: Initialize q = 0.02 and T = 0;
+5: while q < 0.5 and T = 0 do
+6:
+Compute T by Eq. (13);
+7:
+q = q + 0.02;
+8: end while
+9: if T is 0 then
+10:
+Construct clean set via half of the samples with largest
+Wi in Eq. (14) with T = ∞;
+11: else
+12:
+Construct clean set via samples in Eq. (14);
+13: end if
+14: return clean set.
+∥γ2,i(λ)∥2 ̸= 0} and ˜Zi := sup{λ : ∥˜γ2,i(λ)∥2 ̸= 0}. With
+Zi, ˜Zi, we define the Wi as:
+Wi := Zi · sign(Zi − ˜Zi).
+(12)
+Based on these statistics, we define a data-dependent
+threshold T as
+T = max
+�
+t > 0 : 1 + # {j : 0 < Wj ≤ t}
+# {j : −t ≤ Wj < 0} ∨ 1 ≤ q
+�
+,
+(13)
+or T = 0 if this set is empty, where q is the pre-defined upper
+bound. Our algorithm will select the clean subset identified
+by
+C2 := {j : −T ≤ Wj < 0}.
+(14)
+Empirically, T may be equal to 0 if the threshold q is
+sufficiently small. In this regard, no clean data are selected,
+which is meaningless. Therefore, we start with a small q and
+iteratively increase q and calculate T, until an attainable T
+such that T > 0 to bound the FSR as small as possible. In
+practice, when the FSR cannot be bounded by q = 50%,
+we will end the selection and simply select half of the most
+possible clean examples via {Wj}. The whole procedure of
+Knockoffs-SPR is shown in Algorithm 1.
+3.2
+Statistical Analysis about Knockoffs-SPR
+In this part, we present the motivations and intuitions of
+each step in Knockoffs-SPR.
+Data Partition. Knockoffs-SPR partitions the dataset D
+into two subset D1 and D2. This step decomposes the
+dependency of the estimate of β and γ in that we use D1/D2
+to estimate β/γ, respectively. Then ˆβ(D1) is independent of
+ˆγ(D2) if D1 and D2 are disjoint. The independent estimation
+of β and γ makes it provable for FSR control on D2.
+Permutation. As we discussed in step ii, when the original
+label is clean, its permuted label will be a noisy label. On
+the other hand, if the original label is noisy, its permuted
+label changes to clean with probability
+1
+c−1 and noisy with
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+6
+probability
+c−2
+c−1, where c denotes the number of classes.
+Note that γ of noisy data is often selected earlier than that
+of clean data in the solution path. This implies larger Z
+values for noisy data than those for clean data. As a result,
+according to the definition of W, a clean sample will ideally
+have a small negative of W := Z · sign(Z − ˜Z), where Z
+and ˜Z respectively correspond to the clean label and noisy
+label. In contrast for a noisy sample, the W tends to have
+a large magnitude and has approximately equal probability
+to be positive or negative. Such a different behavior of W
+between clean and noisy data can help us to identify clean
+samples from noisy ones.
+Asymmetric comparison statistics W. The classical way to
+define comparison statistics is in a symmetric manner, i.e.,
+Wi := Zi ∨ ˜Zi · sign(Zi − ˜Zi). In this way, a clean sample
+with a noisy permuted label tends to have a large |Wi|, as
+we expect the noisy label to have a large ˜Zi. However, this is
+against our target as we only require clean samples to have
+a small magnitude. For this purpose, we design asymmetric
+comparison statistics that only consider the magnitude of
+the original labels.
+To see the asymmetric behavior of W
+for noisy and
+clean data, we consider the Karush–Kuhn–Tucker (KKT)
+conditions of Eq. (26) with respect to (γ2,i, ˜γ2,i)
+γ2,i + ∂P(γ2,i; λ)
+∂γ2,i
+= x⊤
+2,i(β∗ − ˆβ1) + γ∗
+2,i + ε(2),i,
+(15a)
+˜γ2,i + ∂P(˜γ2,i; λ)
+∂˜γ2,i
+= x⊤
+2,i(β∗ − ˆβ1) + ˜γ∗
+2,i + ˜ε(2),i,
+(15b)
+where ε(2),i ∼i.i.d ˜ε(2),i, |γ∗
+2,i| = |˜γ∗
+2,i| if both y2,i and
+˜y2,i are noisy, and P(γ2,i; λ) := λ|γ2,i| as an example. By
+conditioning on ˆβ1 and denoting ai := x⊤
+2,i(β∗ − ˆβ1), we
+have that
+P(Wi > 0) = P(|ai+γ∗
+2,i+ε2,i| > |ai+ ˜γ∗
+2,i+ ˜ε(2),i|). (16)
+Then it can be seen that if i is clean, we have γ∗
+2,i = 0.
+Then Zi tends to be small and besides, it is probable to have
+Zi < ˜Zi if ˆβ1 can estimate β∗ well. As a result, Wi tends
+to be a small negative. On the other hand, if i is noisy, then
+Zi tends to be large for γi to account for the noisy pattern,
+and besides, it has equal probability between Zi < ˜Zi and
+Zi ≥ ˜Zi when ˜y2,i is switched to another noisy label, with
+probability
+c−2
+c−1. So Wi tends to have a large value and
+besides,
+P(Wi > 0) = P(Wi > 0|˜y2,i is noisy)P(˜y2,i is noisy)
++ P(Wi > 0|˜y2,i is clean)P(˜y2,i is clean) = 1
+2 · c − 2
+c − 1
++ P(Wi > 0|˜y2,i is clean) ·
+1
+c − 1,
+(17)
+which falls in the interval of
+�
+c−2
+c−1 · 1
+2,
+c
+c−1 · 1
+2
+�
+. That is to say,
+P(Wi > 0) ≈ 1
+2. In this regard, the clean data corresponds
+to small negatives of W in the ideal case, which can help
+to discriminate noisy data with large W with almost equal
+probability to be positive or negative.
+Remark. For noisy y2,i, we have P(Wi > 0|˜y2,i is noisy) =
+1/2 by assuming |γ∗
+2,i| = |˜γ∗
+2,i|. However, it may not hold
+in practice when y2,i corresponds to the noisy pattern that
+has been learned by the model. In this regard, it may
+have |γ∗
+2,i| < |˜γ∗
+2,i| for a randomly permuted label ˜y2,i.
+To resolve this problem, we instead set the permutation
+label as the most confident candidate of the model, please
+refer to Sec. 4.1 for details. Besides, if ˆβ1 can accurately
+estimate β∗, according to KKT conditions in Eq. (15), we
+have P(Wi > 0) < 1/2. That is Wi tends to be negative for
+the clean data, which is beneficial for clean sample selection.
+Data-adaptive
+threshold.
+The
+proposed
+data-adaptive
+threshold T
+is directly designed to control the FSR.
+Specifically, the FSR defined in Eq. (9) is equivalent to
+FSR(t) = E
+�# {j : γj ̸= 0 and − t ≤ Wj < 0}
+# {j : −t ≤ Wj < 0} ∨ 1
+�
+,
+(18)
+where the denominator denotes the number of selected
+clean data according to Eq. (14) and the nominator denotes
+the number of falsely selected noisy data. This form of
+Eq. (18) can be further decomposed into,
+E
+� # {γj ̸= 0, −t ≤ Wj < 0}
+1 + # {γj ̸= 0, 0 < Wj ≤ t} · 1 + # {γj ̸= 0, 0 < Wj ≤ t}
+# {−t ≤ Wj < 0} ∨ 1
+�
+≤ E
+� # {γj ̸= 0, −t ≤ Wj < 0}
+1 + # {γj ̸= 0, 0 < Wj ≤ t}
+1 + # {0 < Wj ≤ t}
+# {−t ≤ Wj < 0} ∨ 1
+�
+≤ E
+� # {γj ̸= 0, −t ≤ Wj < 0}
+1 + # {γj ̸= 0, 0 < Wj ≤ t}q
+�
+,
+(19)
+where the last inequality comes from the definition of
+T in Eq. (13). To control the FSR, it suffices to bound
+E
+�
+#{γj̸=0, −t≤Wj<0}
+1+#{γj̸=0, 0<Wj≤t}
+�
+. Roughly speaking, this term means
+the number of negative W to the number of positive
+W, among noisy data. Since W
+for noisy data has
+approximately equal probability to be positive/negative
+as mentioned earlier, intuitively we have this term ≈
+1
+2.
+Formally, we construct a martingale process of 1(Wi > 0)
+among noisy data, which is independent of the magnitude
+|W| due to data partition. We leave these details in the
+appendix.
+3.3
+FSR Control of Knockoffs-SPR
+Our target is to show that FSR ≤ q with our data-adaptive
+threshold T in Eq. (13). Our main result is as follows:
+Theorem 2 (FSR control). For c-class classification task, and for
+all 0 < q ≤ 1, the solution of Knockoffs-SPR holds
+FSR(T) ≤ q
+(20)
+with the threshold T for two subsets defined respectively as
+Ti = max
+�
+t ∈ W : 1 + # {j : 0 < Wj ≤ t}
+# {j : −t ≤ Wj < 0} ∨ 1 ≤ c − 2
+2c q
+�
+.
+We present the proof in the appendix. The coefficient
+1/2 comes from the subset-partition strategy that we run
+Knockoffs-SPR on two D1 and D2, and the term
+c−2
+c
+comes from the upper-bound of the first part in Eq. (19).
+This theorem tells us that FSR can be controlled by the
+given threshold q using the procedure of Knockoffs-SPR.
+Compared to SPR, this procedure is more practical and
+useful in real-world experiments and we demonstrate its
+utility in Sec. 6.3 for more details.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+7
+Algorithm 2 Knockoffs-SPR on full training set
+Input: Noisy feature-label pairs {(xi, yi)}n
+i=1, group class
+size N sample size m, (Optional) clean set.
+Output: clean set.
+1: if Number of classes > N then
+2:
+Compute class prototypes using Eq. (22);
+3:
+Divide classes into groups using Eq. (21);
+4: else
+5:
+Use all classes as a single group;
+6: end if
+7: Construct pieces with uniformly sampled m examples
+for each class (total=N × m);
+8: for each piece do
+9:
+Randomly partition the piece into two sub-pieces A and
+B (each contains Nm/2 examples);
+10:
+Run Algorithm 1 (B, A) on A to get clean-set-A;
+11:
+Run Algorithm 1 (A, B) on B to get clean-set-B;
+12:
+Concat clean-set-A and clean-set-B to get clean-set-piece;
+13: end for
+14: Concat clean-set-pieces to get clean set;
+15: return clean set.
+4
+LEARNING WITH KNOCKOFFS-SPR
+In this section, we introduce how to incorporate Knockoffs-
+SPR into the training of neural networks. We first introduce
+several implementation details of Knockoffs-SPR, then we
+introduce a splitting algorithm that makes Knockoffs-SPR
+scalable to large-scale datasets. Finally, we discuss some
+training strategies to better utilize the selected clean data.
+4.1
+Knockoffs-SPR in Practice
+We introduce several strategies to improve FSR control and
+the power of selecting clean samples, which are inspired by
+different behaviors of W between noisy and clean samples.
+Ideally, for a clean sample i, Wi is expected to be a small
+negative; if i is noisy data, Wi tends to be large and is
+approximately 50% to be positive or negative, as shown
+in Eq. (17). To achieve these properties for better clean
+sample selection, the following strategies are proposed, in
+the procedure of feature extractor, data-preprocessing, label
+permutation strategy, estimating β on D1, and clean data
+identification in Eq. (13), (14).
+Feature Extractor. A good feature extractor is essential for
+clean sample selection algorithms. In our experiments, we
+adopt the self-supervised training method SimSiam [42] to
+pre-train the feature extractor, to make X well encode the
+information of the training data in the early stages.
+Data Preprocessing. We implement PCA on the features
+extracted by neural network for dimension reduction. This
+can make X of full rank, which ensures the identifiability of
+ˆβ in SPR. Besides, such a low dimensionality can make the
+model estimate β more accurately. According to the KKT
+conditions Eq. (15), we have that Wi of clean data i tends
+to be negative with small magnitudes. In this regard, the
+model can have better power of clean sample selection, i.e.,
+selecting more clean samples while controlling FSR.
+Label
+Permutation
+Strategy.
+Instead
+of
+the
+random
+permutation strategy, our Knckoff-SPR permutes the label
+as the most-confident candidate provided by the model at
+each training stage, for FSR consideration especially when
+the noise rate is high or some noisy pattern is dominant
+in the data. Specifically, if the pattern of some noisy label
+y2,i is learned by the model, then γ∗
+2,i may have a smaller
+magnitude than that of ˜γ∗
+2,i for a randomly permuted
+label ˜y2,i that may not be learned by the model, violating
+P(Wi > 0|˜y2,i) = 1/2 and hence P(Wi > 0) ≈ 1/2 in
+practice. In contrast, the most confident permutation can
+alleviate this problem, because the most confident label ˜y2,i
+can naturally have a small magnitude of ˜γ∗
+2,i.
+Estimating β on D1. We implement SPR as the first step
+to learn β on D1. Compared to vanilla OLS, the SPR
+can remove some noisy patterns from data, and hence
+can achieve an accurate estimate of β. Similar to the data
+processing step, such an accurate estimation can improve the
+power of selecting clean samples.
+Clean data identification in Eq. (13), (14). We calculate T
+among W for each class, and identify the clean subset for
+each class, to improve the power of clean data for each
+class. In practice, since some classes may be easier to learn
+than others, the Wi for i in these classes have smaller
+magnitudes. Therefore, data from these classes will take the
+main proportion if we calculate T and identify C2 among all
+classes. With this design, the clean data are more balanced,
+which facilitates the training in the next epochs.
+4.2
+Scalable to Large Dataset
+The computation cost of the sample selection algorithm
+increases with the growth of the training sample, making
+it not scalable to large datasets. To resolve this problem,
+we propose to split the total training set into many pieces,
+each of which contains a small portion of training categories
+with a small number of training data. With the splitting
+strategy, we can run the Knockoffs-SPR on several pieces
+in parallel and significantly reduce the running time. For
+the splitting strategy, we notice that the key to identifying
+clean data is leveraging different behavior in terms of the
+magnitude and the sign of W. Such a difference can be
+alleviated if the patterns from clean classes are similar to the
+noisy ones, which may lead to unsatisfactory recall/power
+of identifying the clean set. This motivates us to group
+similar categories together, to facilitate the discrimination
+of clean data from noisy ones.
+Formally speaking, we define the similarity between the
+class i and j as
+s(i, j) = p⊤
+i pj,
+(21)
+where p represents the class prototype. To obtain pi for the
+class i, we take the clean features xi of each class extracted
+by the network along the training iteration, and average
+them to get the class prototype pc after the current training
+epoch ends, as
+pc =
+�n
+i=1 xi1(yi = c, i ∈ C)
+�n
+i=1 1(yi = c, i ∈ C) ,
+(22)
+Then the most similar classes are grouped together. In
+the initialization step when the clean set has not been
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+8
+Algorithm 3 Training with Knockoffs-SPR.
+Input: Noisy dataset {(imgi, xi, yi)}n
+i=1, p.
+Output: Trained network.
+Initialization :
+1: Model: A self-supervised pre-trained backbone with
+a random initialized fully-connected layer, an EMA
+model;
+2: Initial clean set: Run Algorithm 2 with self-supervised
+pre-trained feature and noisy labels;
+Training Process:
+3: for ep = 0 to max epochs do
+4:
+for each mini-batch do
+5:
+Sample r from U(0, 1);
+6:
+if r > p then
+7:
+Train the network using Eq. (25);
+8:
+else
+9:
+Train the network using Eq. (24);
+10:
+end if
+11:
+Update features x visited in current mini-batch;
+12:
+Update EMA model;
+13:
+end for
+14:
+Run Algorithm 2 on {(xi, yi)}n
+i=1 to get clean set;
+15: end for
+16: return Trained network.
+estimated yet, we simply use all the data to calculate the
+class prototypes. In our experiments, each group is designed
+to have 10 classes.
+For the instances in each group, we split the training data
+of each class in a balanced way such that each piece contains
+the same number of instances for each class. The number
+is determined to ensure that the clean pattern remains the
+majority in the piece, such that optimization can be done
+easily. In practice, we select 75 training data from each
+class to construct the piece. When the class proportion
+is imbalanced in the original dataset, we adopt the over-
+sampling strategy to sample the instance of each class with
+less training data multiple times to ensure that each training
+instance is selected once in some piece. The pipeline of our
+splitting algorithm is described in Algorithm 2.
+4.3
+Network Learning with Knockoffs-SPR
+When training with Knockoffs-SPR, we can further exploit
+the support of noisy data by incorporating Knockoffs-
+SPR with semi-supervised algorithms. In this paper, we
+interpolate part of images between clean data and noisy
+data as in CutMix [38],
+˜
+img = M ⊙ imgclean + (1 − M) ⊙ imgnoisy
+(23a)
+˜y = λyclean + (1 − λ)ynoisy
+(23b)
+where M ∈ {0, 1}W ×H is a binary mask, ⊙ is element-
+wise multiplication, λ ∼ Beta(0.5, 0.5) is the interpolation
+coefficient, and the clean and noisy data are identified
+by Knockoffs-SPR. Then we train the network using the
+interpolated data using
+L
+�
+˜
+img, ˜y
+�
+= LCE
+�
+˜
+img, ˜y
+�
+.
+(24)
+Empirically, we could switch between this semi-supervised
+training with standard supervised training on estimated
+clean data.
+L (imgi, yi) = 1i/∈O · LCE (imgi, yi) ,
+(25)
+where 1i/∈O is the indicator function, which means that
+only the cross-entropy loss of estimated clean data is
+used to calculate the loss. We further store a model with
+EMA-updated weights. Our full algorithm is illustrated in
+Algorithm 3. Neural networks trained with this pipeline
+enjoy powerful recognition capacity in several synthetic and
+real-world noisy datasets.
+5
+RELATED WORK
+Here we make the connections between our Knockoffs-SPR
+and previous research efforts.
+5.1
+Learning with Noisy Labels
+The target of Learning with Noisy Labels (LNL) is to
+train a more robust model from the noisy dataset. We can
+roughly categorize LNL algorithms into two groups: robust
+algorithm and noise detection. A robust algorithm does not
+focus on specific noisy data but designs specific modules
+to ensure that networks can be well-trained even from the
+noisy datasets. Methods following this direction includes
+constructing robust network [2]–[5], robust loss function [6]–
+[9] robust regularization [43]–[46] against noisy labels.
+The noise detection method aims to identify the noisy
+data and design specific strategies to deal with the noisy
+data, including down-weighting the importance in the loss
+function for the network training [47], re-labeling them to
+get correct labels [48], or regarding them as unlabeled data
+in the semi-supervised manner [49], etc.
+For the noise detection algorithm, noisy data are identified
+by some irregular patterns, including large error [14],
+gradient directions [50], disagreement within multiple
+networks [15], inconsistency along the training path [17] and
+some spatial properties in the training data [18], [51]–[53].
+Some algorithms [50], [54] rely on the existence of an extra
+clean set to detect noisy data.
+After detecting the clean data, the simplest strategy is to
+train the network using the clean data only or re-weight
+the data [55] to eliminate the noise. Some algorithms [49],
+[56] regard the detected noisy data as unlabeled data to
+fully exploit the distribution support of the training set in
+the semi-supervised learning manner. There are also some
+studies of designing label-correction module [2], [48], [54],
+[57]–[59] to further pseudo-labeling the noisy data to train
+the network. Few of these approaches are designed from the
+statistical perspective with non-asymptotic guarantees, in
+terms of clean sample selection. In contrast, our Knockoffs-
+SPR can theoretically control the false-selected rate in
+selecting clean samples under general scenarios.
+5.2
+Mean-Shit Parameter
+Mean-shift
+parameters
+or
+incidental
+parameters
+[21]
+originally tackled to solve the robust estimation problem
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+9
+via penalized estimation [60] . With a different focus on
+specific parameters, this formulation address wide attention
+in different research topics, including economics [21]–[24],
+robust regression [20], [25], statistical ranking [26], face
+recognition [27], semi-supervised few-shot learning [28],
+[29], and Bayesian preference learning [30], to name a
+few. Previous work usually uses this formulation to solve
+robust linear models, while in this paper we adopt this to
+select clean data and help the training of neural networks.
+Furthermore, we design an FSR control module and a
+scalable sample selection algorithm based on mean-shift
+parameters with theoretical guarantees.
+5.3
+Knockoffs
+Knockoffs was first proposed in [34] as a data-adaptive
+method to control FDR of variable selection in the sparse
+regression problem. This method was then extended to
+high-dimension regression [1], [61], multi-task regression
+[35], outlier detection [41] and structural sparsity [2]. The
+core of Knockoffs is to construct a fake copy of X as
+negative controls of original features, in order to select true
+positive features with FDR control. Our Knockoffs-SPR is
+inspired by but different from the classical knockoffs in the
+following aspects: i) the Knockoff is to control the FDR,
+i.e., the expected rate of the type-I error while our goal is
+to control the expected rate of the type-II error, a.k.a, FSR,
+in the noisy data scenario; ii) instead of constructing copy
+for X, we construct the copy ˜Y via permutation. Equipped
+with a calibrated data-partitioning strategy, our method can
+control the FER under any desired level.
+6
+EXPERIMENTS
+Datasets. We validate the effectiveness of Knockoffs-
+SPR on synthetic noisy datasets CIFAR-10 and CIFAR-
+100 [62], and real-world noisy datasets WebVision [63]
+and Clothing1M [2]. We consider two types of noisy
+labels for CIFAR: (i) Symmetric noise: Every class is
+corrupted uniformly with all other labels; (ii) Asymmetric
+noise: Labels are corrupted by similar (in pattern) classes.
+WebVision has 2.4 million images collected from the
+internet with the same category list as ImageNet ILSVRC12.
+Clothing1M has 1 million images collected from the internet
+and labeled by the surrounding texts. Thus, the WebVision
+and Clothing1M datasets can be regarded as real-world
+challenges.
+Backbones. For CIFAR, we use ResNet-18 [64] as our
+backbone. For WebVision we use Inception-ResNet [65] to
+extract features to follow previous works. For Clothing1M
+we use ResNet-50 as backbones. For CIFAR and WebVision,
+we respectively self-supervised pretrain for 100 epochs and
+350 epochs using SimSiam [42]. For Clothing1M, we use
+ImageNet pre-trained weights to follow previous works.
+Hyperparameter setting. We use SGD to train all the
+networks with a momentum of 0.9 and a cosine learning
+rate decay strategy. The initial learning rate is set as 0.01.
+The weight decay is set as 1e-4 for Clothing1M, and 5e-
+4 for other datasets. We use a batch size of 128 for all
+experiments. We use random crop and random horizontal
+flip as augmentation strategies. The network is trained
+for 180 epochs for CIFAR, 300 epochs for WebVision, and
+5 epochs for Clothing1M. Network training strategy is
+selected with p = 0.5 (line 6 in Alg. 3) for Clothing1M, while
+for other datasets we only use CutMix training. For features
+used in Knockoffs-SPR, we reduce the dimension of X to
+the number of classes. For Clothing1M, this is 14 and for
+other datasets the reduced dimension is 10 (each piece of
+CIFAR-100 and WebVision contains 10 classes). We also run
+SPR with our new network training algorithm (Alg. 3).
+6.1
+Evaluation on Synthetic Label Noise
+Competitors. We use cross-entropy loss (Standard) as the
+baseline algorithm for two datasets. We compare Knockoffs-
+SPR with algorithms that include Forgetting [66] with
+train the network using dropout strategy, Bootstrap [67]
+which trains with bootstrapping, Forward Correction [55]
+which corrects the loss function to get a robust model,
+Decoupling
+[68]
+which
+uses
+a
+meta-update
+strategy
+to
+decouple
+the
+update
+time
+and
+update
+method,
+MentorNet [12] which uses a teacher network to help train
+the network, Co-teaching [11] which uses two networks to
+teach each other, Co-teaching+ [15] which further uses an
+update by disagreement strategy to improve Co-teaching,
+IterNLD [51] which uses an iterative update strategy,
+RoG [52] which uses generated classifiers, PENCIL [59]
+which uses a probabilistic noise correction strategy, GCE [7]
+and SL [8] which are extensions of the standard cross-
+entropy loss function, and TopoFilter [18] which uses feature
+representation to detect noisy data. For each dataset, all the
+experiments are run with the same backbone to make a
+fair comparison. We randomly run all the experiments five
+times and calculate the average and standard deviation of
+the accuracy of the last epoch. The results of competitors are
+reported in [18].
+As in Table 1, Knockoffs-SPR enjoys a higher performance
+compared with other competitors on CIFAR, validating the
+effectiveness of Knockoffs-SPR on different noise scenarios.
+SPR enjoys better performance on higher symmetric noise
+rate of CIFAR-100. This may contributes to the manual
+selection threshold of 50% of the data. Then SPR will
+select more data than Knockoffs-SPR, for example in Sym.
+80% noise scenario SPR will select 24816 clean data while
+Knockoffs-SPR will select 18185 clean data. This leads to
+a better recovery of clean data (recall of 94.22% while
+Knockoffs-SPR is 81.20%) and thus a better recognition
+capacity.
+6.2
+Evaluation on Real-World Noisy Datasets
+In this part, we compare Knockoffs-SPR with other methods
+on real-world noisy datasets: WebVision and Clothing1M.
+We follow previous work to train and test on the first 50
+classes of WebVision. We also evaluate models trained on
+WebVision to ILSVRC12 to test the cross-dataset accuracy.
+Competitors.
+For
+WebVision,
+we
+compare
+with
+CE
+that
+trains
+with
+cross-entropy
+loss
+(CE),
+as
+well
+as
+Decoupling [68], D2L [69], MentorNet [12], Co-teaching [11],
+Iterative-CV [13], and DivideMix [49]. For clothing1M, we
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+10
+TABLE 1
+Test accuracies(%) on several benchmark datasets with different settings.
+Dataset
+Method
+Sym. Noise Rate
+Asy. Noise Rate
+0.2
+0.4
+0.6
+0.8
+0.2
+0.3
+0.4
+CIFAR-10
+Standard
+85.7 ± 0.5
+81.8 ± 0.6
+73.7 ± 1.1
+42.0 ± 2.8
+88.0 ± 0.3
+86.4 ± 0.4
+84.9 ± 0.7
+Forgetting
+86.0 ± 0.8
+82.1 ± 0.7
+75.5 ± 0.7
+41.3 ± 3.3
+89.5 ± 0.2
+88.2 ± 0.1
+85.0 ± 1.0
+Bootstrap
+86.4 ± 0.6
+82.5 ± 0.1
+75.2 ± 0.8
+42.1 ± 3.3
+88.8 ± 0.5
+87.5 ± 0.5
+85.1 ± 0.3
+Forward
+85.7 ± 0.4
+81.0 ± 0.4
+73.3 ± 1.1
+31.6 ± 4.0
+88.5 ± 0.4
+87.3 ± 0.2
+85.3 ± 0.6
+Decoupling
+87.4 ± 0.3
+83.3 ± 0.4
+73.8 ± 1.0
+36.0 ± 3.2
+89.3 ± 0.3
+88.1 ± 0.4
+85.1 ± 1.0
+MentorNet
+88.1 ± 0.3
+81.4 ± 0.5
+70.4 ± 1.1
+31.3 ± 2.9
+86.3 ± 0.4
+84.8 ± 0.3
+78.7 ± 0.4
+Co-teaching
+89.2 ± 0.3
+86.4 ± 0.4
+79.0 ± 0.2
+22.9 ± 3.5
+90.0 ± 0.2
+88.2 ± 0.1
+78.4 ± 0.7
+Co-teaching+
+89.8 ± 0.2
+86.1 ± 0.2
+74.0 ± 0.2
+17.9 ± 1.1
+89.4 ± 0.2
+87.1 ± 0.5
+71.3 ± 0.8
+IterNLD
+87.9 ± 0.4
+83.7 ± 0.4
+74.1 ± 0.5
+38.0 ± 1.9
+89.3 ± 0.3
+88.8 ± 0.5
+85.0 ± 0.4
+RoG
+89.2 ± 0.3
+83.5 ± 0.4
+77.9 ± 0.6
+29.1 ± 1.8
+89.6 ± 0.4
+88.4 ± 0.5
+86.2 ± 0.6
+PENCIL
+88.2 ± 0.2
+86.6 ± 0.3
+74.3 ± 0.6
+45.3 ± 1.4
+90.2 ± 0.2
+88.3 ± 0.2
+84.5 ± 0.5
+GCE
+88.7 ± 0.3
+84.7 ± 0.4
+76.1 ± 0.3
+41.7 ± 1.0
+88.1 ± 0.3
+86.0 ± 0.4
+81.4 ± 0.6
+SL
+89.2 ± 0.5
+85.3 ± 0.7
+78.0 ± 0.3
+44.4 ± 1.1
+88.7 ± 0.3
+86.3 ± 0.1
+81.4 ± 0.7
+TopoFilter
+90.2 ± 0.2
+87.2 ± 0.4
+80.5 ± 0.4
+45.7 ± 1.0
+90.5 ± 0.2
+89.7 ± 0.3
+87.9 ± 0.2
+SPR
+92.0 ± 0.1
+94.6 ± 0.2
+91.6 ± 0.2
+80.5 ± 0.6
+89.0 ± 0.8
+90.3 ± 0.8
+91.0 ± 0.6
+Knockoffs-SPR
+95.4 ± 0.1
+94.5 ± 0.1
+93.3 ± 0.1
+84.6 ± 0.8
+95.1 ± 0.1
+94.5 ± 0.2
+93.6 ± 0.2
+CIFAR-100
+Standard
+56.5 ± 0.7
+50.4 ± 0.8
+38.7 ± 1.0
+18.4 ± 0.5
+57.3 ± 0.7
+52.2 ± 0.4
+42.3 ± 0.7
+Forgetting
+56.5 ± 0.7
+50.6 ± 0.9
+38.7 ± 1.0
+18.4 ± 0.4
+57.5 ± 1.1
+52.4 ± 0.8
+42.4 ± 0.8
+Bootstrap
+56.2 ± 0.5
+50.8 ± 0.6
+37.7 ± 0.8
+19.0 ± 0.6
+57.1 ± 0.9
+53.0 ± 0.9
+43.0 ± 1.0
+Forward
+56.4 ± 0.4
+49.7 ± 1.3
+38.0 ± 1.5
+12.8 ± 1.3
+56.8 ± 1.0
+52.7 ± 0.5
+42.0 ± 1.0
+Decoupling
+57.8 ± 0.4
+49.9 ± 1.0
+37.8 ± 0.7
+17.0 ± 0.7
+60.2 ± 0.9
+54.9 ± 0.1
+47.2 ± 0.9
+MentorNet
+62.9 ± 1.2
+52.8 ± 0.7
+36.0 ± 1.5
+15.1 ± 0.9
+62.3 ± 1.3
+55.3 ± 0.5
+44.4 ± 1.6
+Co-teaching
+64.8 ± 0.2
+60.3 ± 0.4
+46.8 ± 0.7
+13.3 ± 2.8
+63.6 ± 0.4
+58.3 ± 1.1
+48.9 ± 0.8
+Co-teaching+
+64.2 ± 0.4
+53.1 ± 0.2
+25.3 ± 0.5
+10.1 ± 1.2
+60.9 ± 0.3
+56.8 ± 0.5
+48.6 ± 0.4
+IterNLD
+57.9 ± 0.4
+51.2 ± 0.4
+38.1 ± 0.9
+15.5 ± 0.8
+58.1 ± 0.4
+53.0 ± 0.3
+43.5 ± 0.8
+RoG
+63.1 ± 0.3
+58.2 ± 0.5
+47.4 ± 0.8
+20.0 ± 0.9
+67.1 ± 0.6
+65.6 ± 0.4
+58.8 ± 0.1
+PENCIL
+64.9 ± 0.3
+61.3 ± 0.4
+46.6 ± 0.7
+17.3 ± 0.8
+67.5 ± 0.5
+66.0 ± 0.4
+61.9 ± 0.4
+GCE
+63.6 ± 0.6
+59.8 ± 0.5
+46.5 ± 1.3
+17.0 ± 1.1
+64.8 ± 0.9
+61.4 ± 1.1
+50.4 ± 0.9
+SL
+62.1 ± 0.4
+55.6 ± 0.6
+42.7 ± 0.8
+19.5 ± 0.7
+59.2 ± 0.6
+55.1 ± 0.7
+44.8 ± 0.1
+TopoFilter
+65.6 ± 0.3
+62.0 ± 0.6
+47.7 ± 0.5
+20.7 ± 1.2
+68.0 ± 0.3
+66.7 ± 0.6
+62.4 ± 0.2
+SPR
+72.5 ± 0.2
+75.0 ± 0.1
+70.9 ± 0.3
+38.1 ± 0.8
+71.9 ± 0.2
+72.4 ± 0.3
+70.9 ± 0.5
+Knockoffs-SPR
+77.5 ± 0.2
+74.3 ± 0.2
+67.8 ± 0.4
+30.5 ± 1.0
+77.3 ± 0.4
+76.3 ± 0.3
+73.9 ± 0.6
+TABLE 2
+Test accuracies(%) on WebVision and ILSVRC12 (trained on
+WebVision).
+Method
+WebVision
+ILSVRC12
+top1
+top5
+top1
+top5
+F-correction
+61.12
+82.68
+57.36
+82.36
+Decoupling
+62.54
+84.74
+58.26
+82.26
+D2L
+62.68
+84.00
+57.80
+81.36
+MentorNet
+63.00
+81.40
+57.80
+79.92
+Co-teaching
+63.58
+85.20
+61.48
+84.70
+Iterative-CV
+65.24
+85.34
+61.60
+84.98
+DivideMix
+77.32
+91.64
+75.20
+90.84
+SPR
+77.08
+91.40
+72.32
+90.92
+Knockoffs-SPR
+78.20
+92.36
+74.72
+92.88
+compare with F-correction [55], M-correction [56], Joint-
+Optim [48], Meta-Cleaner [70], Meta-Learning [71], P-
+correction [59], TopoFilter [18] and DivideMix [49].
+The results of real-world datasets are shown in Table 3
+and Table 2, where the results of competitors are reported
+in [49]. Our algorithm Knockoffs-SPR enjoys superior
+performance to almost all the competitors, showing the
+ability of handling real-world challenges. Compared with
+SPR, Knockoffs-SPR also achieves better performance,
+indicating the beneficial of FSR control in real-world
+problems of learning with noisy labels.
+TABLE 3
+Test accuracies(%) on Clothing1M.
+Method
+Accuracy
+Cross-Entropy
+69.21
+F-correction
+69.84
+M-correction
+71.00
+Joint-Optim
+72.16
+Meta-Cleaner
+72.50
+Meta-Learning
+73.47
+P-correction
+73.49
+TopoFiler
+74.10
+DivideMix
+74.76
+SPR
+71.16
+Knockoffs-SPR
+75.25
+6.3
+Evaluation of Sample Selection Quality
+To test whether Knockoffs-SPR leads to better sample
+selection quality, we test the following statistics on CIFAR-
+10 with different noise scenarios, including Sym. 40%, Sym.
+80%, and Asy. 40%. (1) FSR: the ratio of falsely selected
+noisy data in the estimated clean data, which is the target
+that Knockoffs-SPR aims to control; (2) Recall: the ratio of
+selected ground-truth clean data in the full ground-truth
+clean data, which indicates the power of sample selection
+algorithms; (3) F1-score: the harmonic mean of precision (1-
+FSR) and recall, which measures the balanced performance
+of FSR control and power. We plot the corresponding
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+11
+0
+25
+50
+75
+100
+125
+150
+175
+0
+2
+4
+6
+8
+10
+12
+14
+16
+FSR
+Symmetric-40%
+0
+25
+50
+75
+100
+125
+150
+175
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Symmetric-80%
+0
+25
+50
+75
+100
+125
+150
+175
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Asymmetric-40%
+Knockoff-SPR
+Estimated q
+SPR
+TopoFilter
+0
+25
+50
+75
+100
+125
+150
+175
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Recall
+0
+25
+50
+75
+100
+125
+150
+175
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+25
+50
+75
+100
+125
+150
+175
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Training Epochs
+70
+75
+80
+85
+90
+95
+100
+F score
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Training Epochs
+10
+20
+30
+40
+50
+60
+70
+80
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Training Epochs
+65
+70
+75
+80
+85
+90
+95
+100
+Fig. 3. Performance(%) comparison on sample selection along the
+training path on CIFAR 10 with different noise scenarios. In the FSR,
+we also visualize the estimated FSR (q) by Knockoffs-SPR, which is the
+threshold we use to select clean data.
+statistics of each algorithm along the training epochs in
+Fig. 3. We further visualize the estimated FSR, q, of
+Knockoffs-SPR to compare with the ground-truth FSR. As
+we use the splitting algorithm, where each piece contains
+10 classes with each class containing a subset of data, we
+estimate FSR for each piece and report their average and
+standard deviation.
+FSR control in practice. (1) When the noise rate is not
+high, for example in Sym. 40% and Asy. 40% scenarios, the
+ground-truth FSR is well upper-bounded by the estimated
+FSR (with no larger than a single standard deviation). When
+the noise rate is high, for example in Sym. 80% noise
+scenario, the FSR cannot get controlled in the early stage.
+However, as the training goes on, FSR can be well-bounded
+by Knockoffs-SPR.
+(2) When the training set is not very noisy, for example in
+Sym. 40% scenario, the true FSR is far below the estimated
+q. This gap can be explained by a good estimation of
+β due to the small noisy rate. When ˆβ1 can accurately
+estimate β∗, the ˜γ∗
+2,i dominate in Eq. (15). Therefore, the
+P(Wi > 0|˜y2,i is clean) > 1
+2, making P(Wi > 0) > 1/2 >
+c−2
+2(c−1). Since the true FSR bound is inversely proportional to
+P(Wi > 0) (FSR ∝ maxi∈Cc 1/P(Wi > 0) − 1), it is smaller
+than the theoretical bound q.
+Sample selection quality comparison. We compare the
+sample selection quality of Knockoffs-SPR with SPR and
+TopoFilter [18]. (1) Knockoffs-SPR enjoys the (almost) best
+FSR control capacity in all noise scenarios, especially in the
+high noise rate setting. Other algorithms can suffer from
+failure in controlling the FSR (for example in Sym. 80%
+scenario). (2) The power of Knockoffs-SPR is comparable to
+the best algorithms in Sym. 40% and Asy. 40% scenarios. For
+the Sym. 80% case, Knockoffs-SPR sacrifices some power for
+FSR control. (3) Compared together, Knockoffs-SPR enjoys
+the best F1 score on sample selection quality, which well-
+establishes its superiority in selecting clean data with FSR
+control.
+TABLE 4
+Ablation(%) of Knockoffs-SPR on CIFAR-10.
+SPR
+∗-random
+∗-multi
+∗-noPCA
+Knockoffs-SPR
+Sym. 40%
+Acc.
+94.0
+92.0
+94.4
+81.7
+94.7
+FSR
+0.82
+23.04
+1.31
+11.51
+1.27
+q
+-
+4.31±0.73
+2.00±0.00
+14.18±7.62
+5.59±1.11
+Sym. 80%
+Acc.
+78.0
+84.6
+83.0
+10.0
+84.3
+FSR
+60.47
+49.76
+25.77
+78.06
+26.72
+q
+-
+9.47±4.39
+2.22±0.62
+25.95±11.88
+19.52±12.77
+Asy. 40%
+Acc.
+89.5
+84.4
+93.4
+93.7
+93.5
+FSR
+2.19
+16.94
+2.97
+7.62
+2.84
+q
+-
+4.15±2.59
+2.00±0.00
+5.22±2.85
+4.45±2.68
+6.4
+Further Analysis
+Influence
+of
+Knockoffs-SPR
+strategies.
+We
+compare
+Knockoffs-SPR
+with
+several
+variants,
+including:
+SPR
+(The original SPR algorithm), ∗-random (Knockoffs-SPR
+with
+randomly
+permuted
+labels),
+∗-multi
+(Knockoffs-
+SPR without class-specific selection), ∗-noPCA (Knockoffs-
+SPR without using PCA to pre-process the features).
+Experiments are conducted on CIFAR-10 with different
+noise scenarios, as in Table 4. We observe the following
+results:
+(1) As also shown in Fig. 3, the SPR can control the FSR in
+Sym. 40% and Asy. 40% but fails in Sym. 80%. This may
+be due to that when the noisy pattern is not significant,
+the collinearity is weak between noisy samples and clean
+ones, as shown by the distribution of irrepresentable value
+{∥(X⊤
+S XS)−1X⊤
+S Xj∥1}j∈Sc
+in Fig. 1 in the appendix.
+In this regard, most of the earlier (resp. later) selected
+samples in the solution path tend to be noisy (resp. clean)
+samples. When there is strong multi-collinearity and the
+irrepresentable condition is violated seriously, our proposed
+Knockoff procedure can help to control the FSR. The higher
+accuracy of Knockoffs-SPR over SPR can be explained by
+consistent improvements in terms of the F1 score of sample
+selection capacity, as shown in Fig. 3.
+(2) Compared with the random permutation strategy,
+Knockoffs-SPR with most-confident permutation enjoys
+much better FSR control and works much better in Sym.
+40% and Asy. 40% noise scenarios. In Sym. 80% noise
+scenario, the accuracy is comparable, but the most-confident
+permutation still enjoys much better FSR control. This
+result empirically demonstrates the advantage of the most-
+confident permutation over the random permutation.
+(3) Running Knockoffs-SPR on each class separately is
+beneficial for the FSR control capacity and recognition
+capacity. When the noise rate is high, running Knockoffs-
+SPR on multiple classes cannot control the FSR properly by
+the estimated q.
+(4) Using PCA on features as the pre-processing is beneficial
+for FSR control in all cases and will increase the recognition
+capacity in some cases, especially when the noise rate is
+high.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+12
+TABLE 5
+Ablation of the splitting algorithm in computation efficiency (for one
+epoch) on CIFAR-10.
+Model
+Training Time
+Knockoffs-SPR w/o split algorithm
+about 6h
+Knockoffs-SPR w/ split algorithm
+66s
+TABLE 6
+Ablation(%) of training strategies on CIFAR-10.
+Method
+Sym. 40%
+Sym. 80%
+Asy. 40%
+Knockoffs-SPR - Self
+92.5
+24.3
+92.2
+Knockoffs-SPR - Semi
+91.3
+54.0
+88.5
+Knockoffs-SPR - EMA
+94.5
+83.8
+93.2
+Knockoffs-SPR
+94.7
+84.3
+93.5
+Influence of Scalable. In our framework, we propose a split
+algorithm to divide the whole training set into small pieces
+to run Knockoffs-SPR in parallel. In this part, we compare
+the running time between using the split algorithm and not
+using it. Results are shown in Tab. 5. We can see that the
+splitting algorithm can significantly reduce the computation
+time. This is important in large-scale applications.
+Influence of network training strategies. To better train
+the network, we adopt the self-supervised pre-trained
+backbone and the semi-supervised learning framework with
+an EMA update model. In this part, we test the influence of
+these strategies on CIFAR-10 with different noise scenarios.
+Concretely, we compare the full framework with Knockoffs-
+SPR - Self which uses a randomly initialized backbone,
+Knockoffs-SPR - Semi which uses supervised training, and
+Knockoffs-SPR - EMA which does not use the EMA update
+model. Results are summarized in table. 6. We can find
+that: (1) The self-supervised pre-training is important for
+high noise rate scenarios, while for other settings, it is
+not so essential; (2) Semi-supervised training consistently
+improves the recognition capacity, indicating the utility of
+leveraging the support of noisy data; (3) The EMA model
+will slightly improve the recognition capacity.
+airplane
+frog
+automobile
+truck
+bird
+deer
+cat
+dog
+deer
+cat
+dog
+cat
+frog
+deer
+horse
+dog
+ship
+airplane
+truck
+ship
+Fig. 4. Qualitative results of falsely selected examples by Knockoffs-
+SPR. The black words are the labeled classes while the real classes
+are denoted by red words.
+Qualitative visualization. We randomly visualize some
+falsely selected examples of CIFAR-10 in Fig. 4. Most of
+these cases have some patterns that confuse the noisy
+label and the true label, thus making Knockoffs-SPR falsely
+identify them as clean samples.
+7
+CONCLUSION
+This
+paper
+proposes
+a
+statistical
+sample
+selection
+framework – Scalable Penalized Regression with Knockoff
+Filters (Knockoffs-SPR) to identify noisy data with a
+controlled false selection rate. Specifically, we propose an
+equivalent leave-one-out t-test approach as a penalized
+linear model, in which non-zero mean-shift parameters can
+be induced as an indicator for noisy data. We propose a
+delicate Knockoff-SPR algorithm to identify clean samples
+in a way that the false selection rate is controlled by
+the user-provided upper bound. Such an upper bound is
+proved theoretically and works well in empirical results.
+Experiments on several synthetic and real-world datasets
+demonstrate the effectiveness of Knockoff-SPR.
+REFERENCES
+[1]
+C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals,
+“Understanding
+deep
+learning
+requires
+rethinking
+generalization,” in ICLR, 2017. (document)
+[2]
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+massive noisy labeled data for image classification,” in CVPR,
+2015. (document), 5.1, 6
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+Yikai
+Wang
+is
+a
+PhD
+candidate
+at
+the
+School
+of
+Data
+Science,
+Fudan
+University,
+under the supervision of Prof. Yanwei Fu. He
+received a Bachelor’s degree in mathematics
+from the School of Mathematical Sciences,
+Fudan University, in 2019. He published 1
+IEEE TPAMI paper and 2 CVPR papers. His
+current research interests include theoretically
+guaranteed machine learning algorithms and
+applications to computer vision.
+Yanwei Fu received his PhD degree from the
+Queen Mary University of London, in 2014. He
+worked as a post-doctoral researcher at Disney
+Research, Pittsburgh, PA, from 2015 to 2016.
+He is currently a tenure-track professor at Fudan
+University. He was appointed as the Professor
+of Special Appointment (Eastern Scholar) at
+Shanghai Institutions of Higher Learning. He
+published
+more
+than
+80
+journal/conference
+papers including IEEE TPAMI, TMM, ECCV,
+and CVPR. His research interests are one-
+shot/meta-learning, learning-based 3D reconstruction, and image and
+video understanding in general.
+Xinwei Sun is currently an assistant professor
+at the School of Data Science, Fudan University.
+He
+received
+his
+Ph.D.
+in
+the
+school
+of
+mathematical sciences, at Peking University in
+2018. His research interests mainly focus on
+high-dimensional statistics and causal inference,
+with their applications in machine learning and
+medical imaging.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+15
+Supplementary Material
+Yikai Wang, Yanwei Fu, and Xinwei Sun.
+In this supplementary material, we formally present the
+proof of FSR control theorem of knockoff-SPR in Sec. 8.
+For consistency, we also provide the proof of the noisy
+set recovery theorem of SPR in Sec. 9. Some additional
+experimental results are provided in Sec. 10.
+8
+FSR CONTROL THEOREM OF KNOCKOFF-SPR
+Recall that we are solving the problem of
+�
+�
+�
+�
+�
+1
+2
+���Y2 − X2 ˜β1 − γ2
+���
+2
+F + �
+j P(γ2,j; λ),
+1
+2
+��� ˜Y2 − X2 ˜β1 − ˜γ2
+���
+2
+F + �
+j P(˜γ2,j; λ).
+(26)
+We introduce the following lemma from [1] and [2].
+Lemma 3. Suppose that B1, . . . , Bn are indenpendent variables,
+with Bi ∼ Bernoulli(ρi) for each i, where mini ρi ≥ ρ > 0. Let
+J be a stopping time in reverse time with respect to the filtration
+{Fj}, where
+Fj = σ ({B1 + · · · + Bj, Bj+1, . . . , Bn}) .
+Then
+E
+�
+1 + J
+1 + B1 + · · · + BJ
+�
+≤ ρ−1.
+Proof. We first follow [1] to prove the case when {Bi} are
+i.i.d. variables with Bi ∼ Bernoulli(ρ), where ρ > 0. Then
+we follow [2] to generalize the conclusion to non-identical
+case.
+Define the stochastic process
+Mj := 1 + j
+1 + Sj
+with
+Sj := B1 + · · · + Bj
+(27)
+We show that {Mj} is a super-martingale with respect to
+the reverse filtration {Fj}. It is trivial that {Mj} is {Fj}-
+adapted and {Fj} is reverse filtration, that is a decreasing
+sequence
+Fj ⊂ Fj−1 · · · ⊂ {Bi}n
+i=1
+(28)
+with each Fj be a sub-σ-algebras of σ({Bi}n
+i=1). Further,
+we have E [|Mj|] ≤ 1 + j ≤ 1 + n < ∞ with fixed n. Now
+we bound the conditional expectation E[Mj | Fj+1]. Note
+that since {Bj}i+1
+j=1 are i.i.d. variable and thus exchangeable
+when conditioned on Fj+1, then we have
+P(Bj+1 = 1 | Fj+1) = Sj+1
+j + 1
+(29)
+When Sj+1 = 0, it is natural that Sj = 0 thus Mj = 1 + j <
+1 + (j + 1) = Mj+1. When Sj+ > 0, we have
+E[Mj | Fj+1] =
+1 + j
+1 + Sj+1 − 1 · P(Bj+1 = 1 | Fj+1)
++
+1 + j
+1 + Sj+1
+· P(Bj+1 = 0 | Fj+1)
+=1 + j
+Sj+1
+· Sj+1
+j + 1 +
+1 + j
+1 + Sj+1
+· j + 1 − Sj+1
+j + 1
+=1 + (j + 1)
+1 + Sj+1
+=Mj+1.
+(30)
+Hence we have E[Mj | Fj+1] ≤ Mj+1, which finishes the
+proof for the super-martingale {Mj}. Then by the Doob’s
+optional sampling theorem [3], we have
+E[Mj] ≤ E[Mn].
+(31)
+Finally, we have
+E[Mn] = E[ 1 + n
+1 + Sn
+]
+= (1 + n)
+n
+�
+m=0
+1
+1 + m ·
+n!
+m!(n − m)!ρm(1 − ρ)n−m
+= ρ−1(1 − (1 − ρ)n+1)
+≤ ρ−1.
+(32)
+Now it suffices to show that the conclusion also holds for
+non-identical Bernoulli variables. Following [2], for each
+Bi ∼ Bernoulli(ρi), we construct the following disjoint
+Borel sets {Ai
+j}4
+j=1 such that R = ∪4
+j=1Aj with
+P(Ai
+1) = 1 − ρi;
+P(Ai
+2) = ρ1 − ρi
+1 − ρ ;
+P(Ai
+3) = ρρi − ρ
+1 − ρ ;
+P(Ai
+4) = ρi − ρ.
+(33)
+Define Ui = Ai
+1∪Ai
+2, Vi = Ai
+2∪Ai
+3, Gi = Ai
+2∪Ai
+3∪Ai
+4. Based
+on the specific construction we can set Gi = Bi. Further
+define Qi = 1{ξi ∈ Vi} and a random set A = {i : ξi ∈ Ui}.
+Then we have
+Qi · 1{i ∈ A} + 1{i /∈ A}
+= 1{{ξi ∈ Vi ∩ Ui} ∪ {ξi ∈ U C
+i }}
+= 1{{ξi ∈ Ai
+2} ∪ {ξi ∈ Ai
+3 ∪ Ai
+4}}
+= 1{{ξi ∈ Ai
+2 ∪ Ai
+3 ∪ Ai
+4}} = Bi.
+(34)
+Hence
+1 + j
+1 + Sj
+= 1 + |i ≤ j : i ∈ A| + |i ≤ j : i /∈ A|
+1 + �
+i≤j,i∈A Qi + |i ≤ j : i /∈ A|
+≤ 1 + |i ≤ j : i ∈ A|
+1 + �
+i≤j,i∈A Qi
+.
+(35)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+16
+The inequality holds because a+c
+b+c ≤ a
+b for 0 < b ≤ a, c ≥ 0.
+Note that by definition
+P(Qi = 1 | i ∈ A) = P(ξi ∈ Vi | ξi ∈ Ui)
+=
+P(Ai
+2)
+P(Ai
+1 ∪ Ai
+2)
+= ρ = P(Qi = 1),
+P(Qi = 1 | i ̸∈ A) = P(ξi ∈ Vi | ξi /∈ Ui)
+=
+P(Ai
+3)
+P(Ai
+3 ∪ Ai
+4)
+= ρ = P(Qi = 1).
+(36)
+indicating that Qi and A are independent.
+For any fixed A, define ˜Qi = Qi · 1{i ∈ A} and the reverse
+filtration ˜Fj = σ({�j
+k=1 ˜Qk, ˜Qj+1, . . . , ˜Qn, A}). Then when
+conditioned on A, the established result suggests that
+E
+�
+1 + |i ≤ j : i ∈ A|
+1 + �
+i≤j,i∈A Qi
+����A
+�
+≤ ρ−1.
+(37)
+Take expectation over A finishes the proof.
+8.1
+Proof of Theorem 2
+Proof. We first control the FSR rate of the second subset.
+Specifically, we have
+FSR(T) ≤ E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+· 1 + # {j : 0 < Wj ≤ T}
+# {j : −T ≤ Wj < 0} ∨ 1
+�
+≤ q · E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+�
+.
+(38)
+The second inequality holds by the definition of T. Now it
+suffices to show that
+E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+�
+≤
+c
+c − 2.
+(39)
+For γj ̸= 0, we have a probability of
+1
+c−1 to get a clean
+˜γj, leading to Zj > Zj+n with non-zero probability, and
+a probability of c−2
+c−1 to get a noisy ˜γj, where we have no
+information and hence assume a equal probability of Zj >
+Zj+n and Zj < Zj+n. Then we have
+P(Wi > 0) =P(Wi > 0|˜γ∗
+j ̸= 0)P(˜γ∗
+j ̸= 0)
++ P(Wi > 0|˜γ∗
+j = 0)P(˜γ∗
+j = 0)
+≥1
+2 × c − 2
+c − 1 + 0 =
+c − 2
+2(c − 1)
+(40)
+Hence the random variable Bj := 1{Wj>0} ∼ Bernoulli(ρj)
+for γj ̸= 0 with ρj ≥ (c − 2)/(2(c − 1)).
+Now we consider all the Wj of non-null variables, and
+assumes |W1| ≤ · · · ≤ |Wn| with the abuse of subscripts.
+We have
+γj ̸= 0 and − T ≤ Wj < 0
+⇐⇒
+j ≤ J and Bj = 0
+and
+γj ̸= 0 and 0 < Wj ≤ T
+⇐⇒
+j ≤ J and Bj = 1
+Hence
+# {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+= (1 − B1) + · · · + (1 − BJ)
+1 + B1 + · · · + BJ
+=
+1 + J
+1 + B1 + · · · + BJ
+− 1.
+(41)
+If we can use Lemma 3, then
+E
+� # {j : γj ̸= 0 and − T ≤ Wj < 0}
+1 + # {j : γj ̸= 0 and 0 < Wj ≤ T}
+�
+≤ ρ−1−1 ≤
+c
+c − 2
+(42)
+Then we finally get
+FSR(T) ≤ q
+c
+c − 2.
+(43)
+as long as c > 2. Now it suffices to show that our
+random variables {Bj} are mutually independent. This is
+straightforward as we set P(α2; λ) as a sparse penalty for
+each row α2,j in Eq. (26), respectively. Then problem of
+Eq. (26) now is a combination of independent sub-problems
+for each row α2,j, and the solution only depends on
+(x2,j, y2,j, β(λ; D1)). Then with fixed β(λ; D1), the mutual
+independence naturally exist.
+Finally, after we control the FSR rate for the second subset,
+we can get the estimate of β(λ; D2) based on the identified
+clean data in the second subset, and return to run knockoff-
+SPR on the first subset in a similar pipeline. Then we have
+for the whole dataset:
+FSR = E
+�|S1 ∩ C1| + |S2 ∩ C2|
+|C1| + |C2|
+�
+≤ E
+�|S1 ∩ C1|
+|C1|
+�
++ E
+�|S2 ∩ C2|
+|C2|
+�
+≤ 2
+c
+c − 2q.
+(44)
+To control the FSR with q, the threshold of T should be
+defined as c−2
+2c q, which leads to Theorem 2.
+9
+NOISY SET RECOVERY THEOREM OF SPR
+Recall that we are solving the problem of
+min
+⃗γ
+���⃗y − ˚
+X⃗γ
+���
+2
+2 + λ ∥⃗γ∥1 .
+(45)
+Proposition 4. Assume that ˚
+X⊤ ˚
+X is invertible. If
+����λ ˚
+X⊤
+Sc ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1
+ˆvS + ˚
+X⊤
+Sc (I − IS) ( ˚
+Xε)
+����
+∞
+< λ
+(46)
+holds for all ˆvS ∈ [−1, 1]S, where IS = ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ,
+then the estimator ˆ⃗γ of Eq. (45) satisfies that
+ˆS = supp
+�ˆ⃗γ
+�
+⊆ supp (⃗γ∗) = S.
+Moreover, if the sign consistency
+sign
+�ˆ⃗γS
+�
+= sign (⃗γ∗
+S)
+(47)
+holds, Then ˆ⃗γ is the unique solution of (45) with the same sign as
+ˆ⃗γ∗.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+17
+Proof. Note that Eq. (45) is convex that has global minima.
+Denote Eq. (45) as L, the solution of ∂L/∂⃗γ = 0 is the
+unique minimizer. Hence we have
+∂L
+∂⃗γ = − ˚
+X⊤ �
+⃗y − ˚
+X⃗γ
+�
++ λv = 0
+(48)
+where v = ∂ ∥⃗γ∥1 /∂⃗γ. Note that ∥⃗γ∥1 is non-differentiable,
+so we instead compute its sub-gradient. Further note that
+vi = ∂ ∥⃗γ∥1 /∂⃗γi = ∂ |⃗γi| /∂γi. Hence vi = sign (⃗γi) if ⃗γi ̸=
+0 and vi ∈ [−1, 1] if ⃗γi = 0. To distinguish between the two
+cases, we assume vi ∈ (−1, 1) if ⃗γi = 0. Hence there exists
+ˆv ∈ Rn×1 such that
+− ˚
+X⊤ �
+⃗y − ˚
+X ˆ⃗γ
+�
++ λˆv = 0,
+(49)
+ˆvi = sign
+�ˆ⃗γi
+�
+if i ∈ ˆS and ˆvi ∈ (−1, 1) if i ∈ ˆSc.
+To obtain ˆS ⊆ S, we should have ˆ⃗γi = 0 for i ∈ Sc, that is,
+∀i ∈ Sc, |ˆvi| < 1, i.e.,
+��� ˚
+X⊤
+Sc
+�
+⃗y − ˚
+XS ˆ⃗γS
+����
+∞ < λ,
+(50)
+For i ∈ S, we have
+− ˚
+X⊤
+S
+�
+⃗y − ˚
+XS ˆ⃗γS
+�
++ λˆvS = 0.
+(51)
+If ˚
+X⊤ ˚
+X is invertible then
+ˆ⃗γS =
+�
+˚
+X⊤
+S ˚
+XS
+�−1 �
+˚
+X⊤
+S ⃗y − λˆvS
+�
+(52)
+Recall that we have
+⃗y = ˚
+XS⃗γ∗
+S + ˚
+X⃗ε
+(53)
+Hence
+ˆ⃗γS = ⃗γ∗
+S+δS,
+δS :=
+�
+˚
+X⊤
+S ˚
+XS
+�−1 �
+˚
+X⊤
+S ˚
+X⃗ε − λˆvS
+�
+. (54)
+Plugging (54) and (53) into (50) we have
+���� ˚
+X⊤
+Sc ˚
+X⃗ε − ˚
+X⊤
+Sc ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1 �
+˚
+X⊤
+S ˚
+X⃗ε − λˆvS
+�����
+∞
+< λ,
+(55)
+or equivalently
+����λ ˚
+X⊤
+Sc ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1
+ˆvS + ˚
+X⊤
+Sc (I − IS) ˚
+X⃗ε
+����
+∞
+< λ,
+(56)
+where IS
+=
+˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S . To ensure the sign
+consistency, replacing ˆvS
+= sign (⃗γ∗
+S) in the inequality
+above leads to the final result.
+Lemma 5. Assume that ⃗ε is indenpendent sub-Gaussian with
+zero mean and bounded variance Var (⃗εi) ≤ σ2.
+Then with probability at least
+1 − 2cn exp
+�
+�
+�−
+λ2η2
+2σ2 maxi∈Sc
+��� ˚
+Xi
+���
+2
+2
+�
+�
+�
+(57)
+there holds
+��� ˚
+X⊤
+Sc (I − IS)
+�
+˚
+X⃗ε
+����
+∞ ≤ λη
+(58)
+and����
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε
+����
+∞
+≤
+λη
+√Cmin maxi∈Sc
+��� ˚
+Xi
+���
+2
+. (59)
+Proof. Let zc = ˚
+X⊤
+Sc (I − IS)
+�
+˚
+X⃗ε
+�
+, for each i ∈ Sc the
+variance can be bounded by
+Var (zc
+i ) ≤ σ2 ˚
+X⊤
+i (I − IS)2 ˚
+Xi ≤ σ2 max
+i∈Sc
+��� ˚
+Xi
+���
+2
+2 .
+Hoeffding inequality implies that
+P
+���� ˚
+X⊤
+Sc (I − IS)
+�
+˚
+X⃗ε
+����
+∞ ≥ t
+�
+≤ 2 |Sc| exp
+�
+�
+�−
+t2
+2σ2 maxi∈Sc
+��� ˚
+Xi
+���
+2
+2
+�
+�
+� ,
+Setting t = λη leads to the result.
+Now let z =
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε, we have
+Var (z) =
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+XVar (⃗ε) ˚
+X⊤ ˚
+XS
+�
+˚
+X⊤
+S ˚
+XS
+�−1
+≤ σ2 �
+˚
+X⊤
+S ˚
+XS
+�−1
+≤
+σ2
+Cmin
+I.
+Then
+P
+�����
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε
+����
+∞
+≥ t
+�
+≤ 2 |S| exp
+�
+−t2Cmin
+2σ2
+�
+.
+Choose
+t =
+λη
+√Cmin maxi∈Sc
+��� ˚
+Xi
+���
+2
+,
+(60)
+then there holds
+P
+�
+�
+�∥
+�
+˚
+X⊤
+S ˚
+XS
+�−1 ˚
+X⊤
+S ˚
+X⃗ε∥∞ ≥
+λη
+√Cmin maxi∈Sc
+��� ˚
+Xi
+���
+2
+�
+�
+�
+≤ 2 |S| exp
+�
+�
+�−
+λ2η2
+2σ2 maxi∈Sc
+��� ˚
+Xi
+���
+2
+2
+�
+�
+� .
+9.1
+Proof of Theorem 1
+Proof. The proof essentially follows the treatment in [4].
+The results follow by applying Lemma 5 to Proposition 4.
+Inequality (46) holds if condition C2 and the first bound (58)
+hold, which proves the first part of the theorem. The
+sign consistency (47) holds if condition C3 and the second
+bound (59) hold, which gives the second part of the theorem.
+It suffices to show that ˆS ⊆ S implies ˆCc ⊆ Cc. Consider
+one instance i, there are three possible cases for γ∗
+i ∈ R1×c:
+(1) γ∗
+i,j ̸= 0, ∀j ∈ [c]; (2) γ∗
+i,j = 0, ∀j ∈ [c]; (3) ∃j, k ∈
+[c] , s.t. γ∗
+i,j = 0, γ∗
+i,k ̸= 0. If instance i follows case (1) or
+case (3), then i ∈ Cc. If it follows case (2), then i ∈ C, and
+the indexes of all elements of γi are in Sc. Since we have
+ˆS ⊆ S, all elements of γi is in ˆSc, hence i ∈ ˆC. Then we
+have ˆCc ⊆ Cc.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+18
+10
+MORE EXPERIMENTAL RESULTS
+Histogram of the median value of IRR condition of
+SPR. We visualize the median value of the irrepresentable
+(IRR) value, i.e., {∥(X⊤
+S XS)−1X⊤
+S Xj∥1}j of SPR final epoch
+on CIFAR10 with various noisy scenarios in Fig. 5. As
+SPR is running on each piece split from the training set,
+we calculate matrix ˚
+X⊤
+Sc ˚
+XS( ˚
+X⊤
+S ˚
+XS)−1 in irrepresentable
+condition (C2 in Theorem 1) for each piece at the final
+epoch. Then the L1 norm of each row of the matrix is the
+IRR value of corresponding clean data. The median value
+of IRR values in a single piece is used to construct the
+histogram. For the noise scenario of Asy. 40% and Sym. 40%,
+the median IRR value is small, indicating weak collinearity
+between clean data and noisy data. In these cases, SPR
+has more chance to distinguish noisy data from clean data
+and thus leads to a good FSR control capacity. For the
+noise scenario of Sym. 80%, the median IRR values are
+much larger, indicating a strong multi-collinearity. Thus SPR
+can hardly distinguish between clean data and noisy data,
+leading to a high FSR rate.
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Symmetric-40%
+4
+6
+8
+10
+12
+14
+16
+0
+5
+10
+15
+20
+25
+Symmetric-80%
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0
+10
+20
+30
+40
+50
+60
+Asymmetric-40%
+Fig. 5. Histogram of the median value of the IRR value of SPR on
+CIFAR10 with various noisy scenarios.
+REFERENCES
+[1] Rina Foygel Barber and Emmanuel J. Cand‘es. A knockoff filter
+for high-dimensional selective inference. The Annals of Statistics,
+47(5):2504 – 2537, 20 (document), 3.1, 5.3, 8, 8
+[2] Yang Cao, Xinwei Sun, and Yuan Yao. Controlling the false
+discovery rate in transformational sparsity: Split knockoffs. In
+arXiv, 2021. (document), 3.1, 5.3, 8, 8, 8
+[3] Joseph L Doob. Stochastic processes. Wiley New York, 195 8
+[4] M.
+J.
+Wainwright,
+“Sharp
+thresholds
+for
+high-dimensional
+and noisy sparsity recovery using l1 -constrained quadratic
+programming (lasso),” IEEE transactions on information theory,
+2009. (document), 2.3, 9.1
+

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@@ -0,0 +1,1611 @@
+Generic transversality of radially symmetric
+stationary solutions stable at infinity for
+parabolic gradient systems
+Emmanuel Risler
+January 9, 2023
+This paper is devoted to the generic transversality of radially symmetric
+stationary solutions of nonlinear parabolic systems of the form
+∂tw(x, t) = −∇V
+�w((x, t))
+� + ∆xw(x, t) ,
+where the space variable x is multidimensional and unbounded. It is proved
+that, generically with respect to the potential V , radially symmetric stationary
+solutions that are stable at infinity (in other words, that approach a minimum
+point of V at infinity in space) are transverse; as a consequence, the set of
+such solutions is discrete. This result can be viewed as the extension to
+higher space dimensions of the generic elementarity of symmetric standing
+pulses, proved in a companion paper. It justifies the generic character of the
+discreteness hypothesis concerning this set of stationary solutions, made in
+another companion paper devoted to the global behaviour of (time dependent)
+radially symmetric solutions stable at infinity for such systems.
+2020 Mathematics Subject Classification: 35K57, 37C20, 37C29.
+Key words and phrases: parabolic gradient systems, radially symmetric stationary solutions, generic
+transversality, Morse–Smale theorem.
+1
+arXiv:2301.02605v1  [math.AP]  6 Jan 2023
+
+Contents
+1
+Introduction
+3
+1.1
+An insight into the main result . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.2
+Radially symmetric stationary solutions stable at infinity
+. . . . . . . . .
+3
+1.3
+Differential systems governing radially symmetric stationary solutions
+. .
+4
+1.4
+Transversality of radially symmetric stationary solutions stable at infinity
+7
+1.5
+The space of potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.6
+Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.7
+Key differences with the generic transversality of standing pulses in space
+dimension one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2
+Preliminary properties
+10
+2.1
+Proof of Lemma 1.4
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2.2
+Transversality of homogeneous radially symmetric stationary solutions
+stable at infinity
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+2.3
+Additional properties close to the origin . . . . . . . . . . . . . . . . . . .
+13
+2.4
+Additional properties close to infinity . . . . . . . . . . . . . . . . . . . . .
+14
+3
+Tools for genericity
+15
+4
+Generic transversality among potentials that are quadratic past a given radius 17
+4.1
+Notation and statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+4.2
+Reduction to a local statement
+. . . . . . . . . . . . . . . . . . . . . . . .
+17
+4.3
+Proof of the local statement (Proposition 4.2) . . . . . . . . . . . . . . . .
+18
+4.3.1
+Setting
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+4.3.2
+Equivalent characterizations of transversality . . . . . . . . . . . .
+19
+4.3.3
+Checking hypothesis 1 of Theorem 4.2 of [1] . . . . . . . . . . . . .
+20
+4.3.4
+Checking hypothesis 2 of Theorem 4.2 of [1] . . . . . . . . . . . . .
+21
+4.3.5
+Conclusion
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+5
+Proof of the main results
+24
+2
+
+1 Introduction
+1.1 An insight into the main result
+The purpose of this paper is to prove the generic transversality of radially symmetric
+stationary solutions stable at infinity for gradient systems of the form
+(1.1)
+∂tw(x, t) = −∇V
+�w((x, t))
+� + ∆xw(x, t) ,
+where time variable t is real, space variable x lies in the spatial domain Rdsp with dsp an
+integer not smaller than 2, the state function (x, t) �→ w(x, t) takes its values in Rdst with
+dst a positive integer, and the nonlinearity is the gradient of a scalar potential function
+V : Rdst → R, which is assumed to be regular (of class at least C2). An insight into the
+main result of this paper (Theorem 1 on page 8) is provided by the following corollary.
+Corollary 1.1. For a generic potential V , the following conclusions hold:
+1. every radially symmetric stationary solution stable at infinity of system (1.1) is
+robust with respect to small perturbations of V ;
+2. the set of all such solutions is discrete.
+The discreteness stated in conclusion 2 of this corollary is a required assumption for
+the main result of [4], which describes the global behaviour of radially symmetric (time
+dependent) solutions stable at infinity for the parabolic system (1.1). Corollary 1.1
+provides a rigorous proof that this assumption holds generically with respect to V .
+This paper can be viewed as a supplement of the article [1], which is devoted to the
+generic transversality of bistable travelling fronts and standing pulses stable at infinity
+for parabolic systems of the form (1.1) in (unbounded) space dimension one, and which
+provides a rigorous proof of the genericity of similar assumptions made in [2, 3, 5]. The
+ideas, the nature of the results, and the scheme of the proof are the same.
+1.2 Radially symmetric stationary solutions stable at infinity
+A function u : [0, +∞) → Rdst, r �→ u(r) defines a radially symmetric stationary solution
+of the parabolic system (1.1) if and only if it satisfies, on (0, +∞), the (non-autonomous)
+differential system
+(1.2)
+¨u(r) = −dsp − 1
+r
+˙u(r) + ∇V
+�u(r)
+� ,
+where ˙u and ¨u stand for the first and second derivatives of r �→ u(r), together with the
+limit
+(1.3)
+˙u(r) → 0
+as
+r → 0+ .
+Observe that, in this case, u(·) is actually the restriction to [0, +∞) of an even function in
+C3(R, Rd
+st) which is a solution (on R) of the differential system (1.2) (the limit (1.3) ensures
+3
+
+that equality (1.2) still makes sense and holds at r equals 0). In other words, provided
+that condition (1.3) holds, it is equivalent to assume that system (1.2) holds on (0, +∞)
+or on [0, +∞). By abuse of language, the terminology radially symmetric stationary
+solution of system (1.1) will refer, all along the paper, to functions u : [0, +∞) → Rdst
+satisfying these conditions (1.2) and (1.3) (even if, formally, it is rather the function
+Rd
+sp → Rd
+st, x �→ u
+�|x|
+� that fits with this terminology).
+Let us denote by Σmin(V ) the set of nondegenerate (local or global) minimum points
+of V ; with symbols,
+Σmin(V ) =
+�u ∈ Rdst : ∇V (u) = 0 and D2V (u) > 0
+� .
+Throughout all the paper, the words minimum point will be used to denote a local or
+global minimum point of a (potential) function.
+Definition 1.2. A (global) solution (0, +∞) → Rdst, r �→ u(r), of the differential system
+(1.2) (in particular a radially symmetric stationary solution of system (1.1)) is said to be
+stable at infinity if u(r) approaches a point of Σmin(V ) as r goes to +∞. If this point of
+Σmin(V ) is denoted by u∞, then the solution is said to be stable close to u∞ at infinity.
+Notation. For every u∞ in Σmin(V ), let SV, u∞ denote the set of the radially symmetric
+stationary solutions of system (1.1) that are stable close to u∞ at infinity. With symbols,
+SV, u∞ =
+�u : [0, +∞) → Rdst : u satisfies (1.2) and (1.3) and u(r) −−−−→
+r→+∞ u∞
+� .
+Let
+S0
+V, u∞ =
+�
+u(0) : u ∈ SV, u∞
+�
+,
+and let
+(1.4)
+SV =
+�
+u∞∈Σmin(V )
+SV, u∞
+and
+S0
+V =
+�
+u∞∈Σmin(V )
+S0
+V, u∞ .
+The following statement is an equivalent (simpler) formulation of conclusion 2 of
+Corollary 1.1.
+Corollary 1.3. For a generic potential V , the subset S0
+V of Rdst is discrete.
+1.3 Differential systems governing radially symmetric stationary solutions
+The second-order differential system (1.2) is equivalent to the (non-autonomous) 2dst-
+dimensional first order differential differential system
+(1.5)
+�
+�
+�
+�
+�
+˙u = v
+˙v = −dsp − 1
+r
+v + ∇V (u) .
+Introducing the auxiliary variables τ and c defined as
+(1.6)
+τ = log(r)
+and
+c = 1
+r ,
+4
+
+the previous 2dst-dimensional differential system (1.5) is equivalent to each of the following
+two 2dst + 1-dimensional autonomous differential systems:
+(1.7)
+�
+�
+�
+�
+�
+�
+�
+uτ = rv
+vτ = −(dsp − 1)v + r∇V (u)
+rτ = r ,
+and
+(1.8)
+�
+�
+�
+�
+�
+�
+�
+ur = v
+vr = −(dsp − 1)cv + ∇V (u)
+cr = −c2 .
+Remark. Integrating the third equations of systems (1.7) and (1.8) yields
+r = r0eτ−τ0
+and
+1
+c − 1
+c0
+= r − r0 ,
+and the parameters τ0 and c0 (which determine in each case the origin of “time”) do
+not matter in principle, since those systems are autonomous. However, if the “initial
+conditions” r0 and c0 are positive (which is true for the solutions that describe radially
+symmetric stationary solutions of system (1.1)), it is natural to choose, in each case, the
+origins of time according to equalities (1.6), that is :
+τ0 = ln(r0)
+and
+c0 = 1
+r0
+.
+Properties close to origin.
+System (1.7) is relevant to provide an insight into the limit
+system (1.5) as r goes to 0. The subspace R2dst × {0} (r equal to 0) is invariant by the
+flow of this system, and the system reduces on this invariant subspace to
+(1.9)
+�
+uτ = 0
+vτ = −(dsp − 1)v ,
+see figure 1.1. For every u0 in Rdst, the point (u0, 0Rdst, 0) is an equilibrium of sys-
+tem (1.7); let us denote by W u, 0
+V
+(u0) the (one-dimensional) unstable manifold of this
+equilibrium, for this system, let
+(1.10)
+W u, 0, +
+V
+(u0) = W u, 0
+V
+(u0) ∩
+�R2dst × (0, +∞)
+� ,
+and let
+W u, 0, +
+V
+=
+�
+u0∈Rdst
+W u, 0, +
+V
+(u0) .
+The subspace
+(1.11)
+Ssym = Rdst × {0Rdst} × {0}
+of R2dst+1 can be seen as the higher space dimension analogue of the symmetry (reversibil-
+ity) subspace Rdst × {0Rdst} of R2dst (which is relevant for symmetric standing pulses in
+space dimension 1, see [1] and subsection 1.7 below); the set W u, 0, +
+V
+can be seen as the
+unstable manifold of this subspace Ssym.
+5
+
+Figure 1.1: Dynamics of the (equivalent) differential systems (1.7) (for r nonnegative
+finite) and (1.8) (for c = 1/r nonnegative finite) in Rdst × Rdst × [0, +∞] (this domain is
+three-dimensional if dst is equal to 1, as on the figure). For the limit differential system
+(1.9) in the subspace r = 0 (in green), the trajectories are vertical and the solutions
+converge towards the horizontal u-axis, defined as Ssym in (1.11), and which is the higher
+space dimensional analogue of the symmetry subspace for symmetric standing pulses
+in space dimension 1. The point u∞ is a local minimum point of V , so that the point
+(u∞, 0Rdst) is a hyperbolic equilibrium for the limit differential system (1.12) in the
+subspace c = 0 ⇐⇒ r = +∞ (in blue). Systems (1.7) and (1.8) are autonomous, but the
+quantity r (the quantity c) goes monotonously from 0 to +∞ (from +∞ to 0) for all the
+solutions in the subspace r > 0 ⇐⇒ c > 0, so that those solutions can be parametrized
+with r (with c) as time. The unstable manifold W u, 0, +
+V
+(u0) is one-dimensional and is a
+transverse intersection between the unstable set W u, 0, +
+V
+of the subspace {r = 0, v = 0Rdst}
+and the centre stable manifold W cs, ∞, +
+V
+(u∞) of the equilibrium (u∞, 0Rdst, c = 0). To
+prove the generic transversality of this intersection is the main goal of the paper. The
+dotted red curve is the projection onto the (u, r)-subspace of this intersection. The part of
+W cs, ∞, +
+V
+(u∞) which is displayed on the figure can also be seen as the local centre stable
+manifold W cs, ∞, +
+loc, V, ε1, c1(u∞) defined in (2.10) (with u∞ equal to the point u∞,1 introduced
+there).
+6
+
+wo
+u
+sym
+om
+L
+0
+(8m)
+loc, V, E1, C1
+C1
+L0=
+u
+8
+E1Properties close to infinity.
+System (1.8) is relevant to provide an insight into the limit
+system (1.5) as r goes to +∞. The subspace R2dst × {0} of R2dst+1 (c equal to 0, or
+in other words r equal to +∞) is invariant by the flow of this system, and the system
+reduces on this invariant subspace to
+(1.12)
+�
+ur = v
+vr = ∇V (u) .
+For every u∞ in Σmin(V ), the point (u∞, 0Rdst, 0) is an equilibrium of system (1.8); let
+us consider its global centre-stable manifold in R2dst × (0, +∞), defined as
+(1.13)
+W cs, ∞, +
+V
+(u∞) =
+�
+(u0, v0, c0) ∈ R2dst × (0, +∞) : the solution of system (1.8)
+with initial condition (u0, v0, c0) at “time” r0 = 1/c0 is
+defined up to +∞ and goes to (u∞, 0, 0) as r goes to +∞
+�
+.
+This set W cs, ∞, +
+V
+(u∞) is a dst + 1-dimensional submanifold of R2dst × (0, +∞) (see
+subsection 2.4).
+Radially symmetric stationary solutions.
+Let us consider the involution
+ι : R2dst × (0, +∞) → R2dst × (0, +∞) ,
+(u, v, r) �→ (u, v, 1/r) .
+The following lemma, proved in subsection 2.1, formalizes the correspondence between
+the radially symmetric stationary solutions stable at infinity for system (1.1) and the
+manifolds defined above.
+Lemma 1.4. Let u∞ be a point of Σmin(V ). A (global) solution [0, +∞) → Rdst, r �→ u(r)
+of system (1.2) belongs to SV, u∞ if and only if its trajectory (in R2dst × (0, +∞))
+(1.14)
+��u(r), ˙u(r), r
+� : r ∈ (0, +∞)
+�
+belongs to the intersection
+(1.15)
+W u, 0, +
+V
+∩ ι−1�W cs, ∞, +
+V
+(u∞)
+� .
+1.4 Transversality of radially symmetric stationary solutions stable at infinity
+Definition 1.5. Let u∞ be a point of Σmin(V ). A radially symmetric stationary solution
+stable close to u∞ at infinity for system (1.1) (in other words, a function u of SV, u∞) is
+said to be transverse if the intersection (1.15) is transverse, in R2dst × (0, +∞), along the
+trajectory (1.14).
+Remark. The natural analogue of radially symmetric stationary solutions stable at infinity
+when space dimension dsp is equal to 1 are symmetric standing pulses stable at infinity
+(see Definition 1.5 of [1]), and the natural analogue for such pulses of Definition 1.5 above
+7
+
+is their elementarity, not their transversality (see Definition 1.4 and Definition 1.6 of [1]).
+However, the transversality of a symmetric standing pulse (when the space dimension
+dsp equals 1) makes little sense in higher space dimension, because of the singularity at r
+equals 0 for the differential systems (1.2) and (1.5), or because of the related fact that
+the subspace {r = 0} is invariant for the differential system (1.7). For that reason, the
+adjective transverse (not elementary) is chosen to qualify the property considered in
+Definition 1.5 above.
+1.5 The space of potentials
+For the remaining of the paper, let us take and fix an integer k not smaller than 1. Let us
+consider the space Ck+1
+b
+(Rdst, R) of functions Rd → R of class Ck+1 which are bounded,
+as well as their derivatives of order not larger than k + 1, equipped with the norm
+∥W∥Ck+1
+b
+=
+max
+α multi-index, |α|≤k+1 ∥∂|α|
+uαW∥L∞(Rd,R) ,
+and let us embed the larger space Ck+1(Rdst, R) with the following topology: for V in
+this space, a basis of neighbourhoods of V is given by the sets V + O, where O is an
+open subset of Ck+1
+b
+(Rdst, R) embedded with the topology defined by ∥·∥Ck+1
+b
+(which can
+be viewed as an extended metric). For comments concerning the choice of this topology,
+see subsection 1.4 of [1].
+1.6 Main result
+The following generic transversality statement is the main result of this paper.
+Theorem 1 (generic transversality of radially symmetric stationary solutions stable at
+infinity). There exists a generic subset G of
+�
+Ck+1(Rdst, R), ∥·∥Ck+1
+b
+�
+such that, for every
+potential function V in G, every radially symmetric stationary solution stable at infinity
+of the parabolic system (1.1) is transverse.
+Theorem 1 can be viewed as the extension to higher space dimensions (for radially
+symmetric solutions) of conclusion 2 of Theorem 1.7 of [1] (which is concerned with
+elementary standing pulses stable at infinity in space dimension 1). A short comparison
+between these two results and their proofs is provided in the next subsection. For more
+comments and a short historical review on transversality results in similar contexts, see
+subsection 1.6 of the same reference.
+The core of the paper (section 4) is devoted to the proof of the conclusions of Theorem 1
+among potentials which are quadratic past a certain radius (defined in (3.2)), as stated
+in Proposition 4.1. The extension to general potentials of Ck+1
+b
+(Rdst, R) is carried out in
+section 5.
+Remark. As in [1] (see Theorem 1.8 of that reference), the same arguments could be
+called upon to prove that the following additional conclusions hold, generically with
+respect to the potential V :
+8
+
+1. for every minimum point of V , the smallest eigenvalue of D2V at this minimum
+point is simple;
+2. every radially symmetric stationary solution stable at infinity of the parabolic system
+(1.1) approaches its limit at infinity tangentially to the eigenspace corresponding to
+the smallest eigenvalue of D2V at this point.
+1.7 Key differences with the generic transversality of standing pulses in
+space dimension one
+Table 1.1 lists the key differences between the proof of the generic elementarity of
+symmetric standing pulses carried out in [1], and the proof of the generic transversality
+of radially symmetric stationary solutions carried out in the present paper (implicitly,
+the other steps/features of the proofs are similar or identical). The state dimension,
+which is simply denoted by d in [1], is here denoted by dst in both cases. Some of the
+notation/rigour is lightened.
+Symmetric standing pulse
+Radially symmetric
+stationary solution
+Critical point at infinity
+critical point e, E = (e, 0Rdst )
+minimum point u∞
+Symmetry subspace Ssym
+{(u, v) ∈ R2dst : v = 0},
+dimension dst
+{(u, v, r) ∈ R2dst+1 : (v, r) = (0, 0)},
+dimension dst
+Differential system
+governing the profiles
+autonomous, conservative,
+regular at Ssym
+non-autonomous, dissipative,
+singular at reversibility subspace
+Direction of the flow
+E → Ssym
+Ssym → u∞
+Invariant manifold at
+infinity
+W u(E), dimension dst − m(e)
+W cs, ∞, +(u∞), dimension dst + 1
+Invariant manifold at
+symmetry subspace
+none
+W u, 0, +, dimension dst + 1
+Transversality
+W u(E) ⋔ Ssym
+W cs, ∞, +(u∞) ⋔ W u, 0, +
+Transversality of spatially
+homogeneous solutions
+irrelevant
+Proposition 2.2
+Interval Ionce (values
+reached only once)
+“anywhere”
+close to Ssym
+M (departure set of Φ)
+parametrization of ∂W u
+loc, V (E)
+and time, dimension dst − m(e)
+Ssym and W cs, ∞, +
+loc
+(u∞) at r = N,
+dimension 2dst
+N (arrival set of Φ)
+R2dst
+R2dst × R2dst
+W (target manifold)
+Ssym
+diagonal of N
+dim(M) − codim(W)
+−m(e)
+0
+Condition to be fulfilled
+by perturbation W
+�
+DΦ(W)
+�� (0, ψ)�
+̸= 0
+�
+DΦu(W)
+�� (φ, ψ)�
+̸= 0
+Perturbation W, case 3
+precluded
+W(u0) ̸= 0
+Table 1.1: Formal comparison between the generic elementarity of symmetric standing
+pulses (space dimension 1) proved in [1], and the generic transversality of radially
+symmetric stationary solutions (higher space dimension dsp) proved in the present paper.
+Here are a few additional comments about these differences.
+9
+
+Concerning the critical point at infinity, u∞ is assumed (here) to be a minimum point,
+whereas (in [1]) the Morse index of e is any. Indeed, if the Morse index m(u∞) of u∞ was
+positive, then the dimension of the centre-stable manifold W cs, ∞, +
+V
+(u∞) would be equal
+to dst + m(u∞) + 1; as a consequence, proving the transversality of the intersection (1.15)
+in that case would require more stringent regularity assumptions on V (see hypothesis
+1 of Theorem 4.2 of [1]) while nothing particularly useful could be derived from this
+transversality. On the other hand, assuming that u∞ is a minimum point allows to view
+its local centre-stable manifold as a graph (u, c) �→ v (see Proposition 2.4), which is
+slightly simpler.
+Concerning the interval Ionce providing values u reached “only once” by the profile
+(Lemma 2.3), the proof of the present paper takes advantage of the dissipation to find a
+convenient interval close to the “departure point” u0, as was done in [1] for travelling
+fronts (whereas, for standing pulse, the interval is to be found “anywhere”, thanks to the
+conservative nature of the differential system governing the profiles, see conclusion 1 of
+Proposition 3.3 of [1]).
+Concerning the function Φ to which Sard–Smale theorem is applied in the present
+paper, both manifolds W u, 0, + and W cs, ∞, +(u∞) depend on the potential V . However,
+the transversality of an intersection between these two manifolds can be seen as the
+transversality of the image of Φ with the (fixed) diagonal of R2dst × R2dst, for a function
+Φ combining the parametrization of these two manifolds. This trick, which is the same
+as in [1] for travelling fronts, allows to apply Sard–Smale theorem to a function Φ with a
+fixed arrival space N containing a fixed target manifold W (in this case the diagonal of
+N). By contrast, for symmetric standing pulses in [1], since the subspace Ssym involved
+in the transverse intersection is fixed, the previous trick is unnecessary and the setting is
+simpler.
+Finally, a technical difference occurs in “case 3” of the proof that the degrees of freedom
+provided by perturbing the potential allow to reach enough directions in the arrival state
+of Φ (Lemma 4.6, which is the core of the proof). In [1], case 3 is shown to lead to a
+contradiction, not only for symmetric standing pulses, but also for asymmetric ones and
+for travelling fronts. Here, such a contradiction does not seem to occur (or at least is
+more difficult to prove), but this has no harmful consequence: a suitable perturbation of
+the potential can still be found in this case.
+2 Preliminary properties
+2.1 Proof of Lemma 1.4
+Let V denote a potential function in Ck+1(Rdst, R). Let (0, +∞) → Rdst, r �→ u(r) denote
+a (global) solution of system (1.2), assumed to be stable close to some point u∞ of
+Σmin(V ) at infinity (Definition 1.2). Lemma 1.4 follows from the next lemma.
+Lemma 2.1. The derivative ˙u(r) goes to 0 as r goes to +∞.
+10
+
+Proof. Let us consider the Hamiltonian function
+(2.1)
+HV : R2dst → R ,
+(u, v) �→ v2
+2 − V (u) ,
+and, for every r in (0, +∞), let
+h(r) = HV
+�u(r), ˙u(r)
+� .
+It follows from system (1.2) that, for every r in (0, +∞),
+(2.2)
+˙h(r) = −dsp − 1
+r
+˙u(r)2 ,
+thus the function h(·) decreases, and it follows from the expression (2.1) of the Hamiltonian
+that this function converges, as r goes to +∞, towards a finite limit h∞ which is not
+smaller than −V (u∞).
+Let us proceed by contradiction and assume that h∞ is larger than −V (u∞). Then, it
+follows again from the expression (2.1) of the Hamiltonian that the quantity ˙u(r)2 con-
+verges towards the positive quantity 2
+�h∞ + V (u∞)
+� as r goes to +∞. As a consequence,
+it follows from equality (2.2) that h(r) goes to −∞ as r goes to +∞, a contradiction.
+Lemma 2.1 is proved.
+2.2 Transversality of homogeneous radially symmetric stationary solutions
+stable at infinity
+Proposition 2.2. For every potential function V in Ck+1(Rdst, R) and for every nonde-
+generate minimum point u∞ of V , the constant function
+[0, +∞) → Rdst ,
+r �→ u∞ ,
+which defines an (homogeneous) radially symmetric stationary solution stable at infinity
+for system (1.1) , is transverse (in the sense of Definition 1.5).
+Proof. Let V denote a function in Ck+1(Rdst, R) and u∞ denote a nondegenerate minimum
+point of V . The function [0, +∞) → Rdst, r �→ u∞ is a (constant) solution of the
+differential system (1.5), and the linearization of this differential system around this
+solution reads
+(2.3)
+¨u = −dsp − 1
+r
+˙u + D2V (u∞) · u .
+Let (0, +∞) → Rdst, r �→ u(r) denote a nonzero solution of this differential system, and,
+for every r in (0, +∞), let
+v(r) = ˙u(r)
+and
+U(r) =
+�u(r), v(r)
+�
+and
+q(r) = u(r)2
+2
+.
+11
+
+Then (omitting the dependency on r),
+˙q = u · ˙u
+and
+¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1
+r
+˙q + D2V (u∞) · (u, u) ,
+so that
+d
+dr
+�rdsp−1 ˙q(r)
+� = rdsp−1
+�
+¨q + dsp − 1
+r
+˙q
+�
+= rdsp−1� ˙u2 + D2V (u∞) · (u, u)
+� .
+Since r �→ u(r) was assumed to be nonzero, it follows that the quantity rdsp−1 ˙q(r) is
+strictly increasing on (0, +∞). To prove the intended conclusion, let us proceed by
+contradiction and assume that, for every r in (0, +∞),
+�u(r), v(r), r
+� belongs:
+1. to the tangent space T(u∞,0Rdst ,r)W u, 0, +
+V
+(u∞),
+2. and to the tangent space T(u∞,0Rdst ,r)
+�
+ι−1�W cs, ∞, +
+V
+(u∞)
+��
+.
+As in (1.6), let us introduce the auxiliary variables τ (equal to log(r)) and c (equal to
+1/r). With this notation, system (2.3) is equivalent to
+(2.4)
+�
+�
+�
+�
+�
+�
+�
+uτ = rv
+vτ = −(dsp − 1)v + rD2V (u∞) · u
+rτ = r ,
+and to
+(2.5)
+�
+�
+�
+�
+�
+�
+�
+ur = v
+vr = −(dsp − 1)cv + D2V (u∞) · u
+cr = −c2 .
+Assumptions 1 and 2 above yield the following conclusions.
+1. In view of the limit of system (2.4) as r goes to 0+, it follows from assumption 1
+that there exists δu0 in Rdst such that
+�u(r), v(r)
+� goes to (δu0, 0Rdst) as r goes to
+0+;
+2. and in view of the limit of system (2.5) as c goes to 0+, it follows from assumption
+2 that
+�u(r), v(r)
+� goes to (0Rdst, 0Rdst), at an exponential rate, as r goes to +∞.
+It follows from these two conclusions that the quantity rdsp−1 ˙q(r) goes to 0 as r goes
+to 0+ and as r goes to +∞, a contradiction with the fact (observed above) that this
+quantity is strictly increasing with r. Proposition 2.2 is proved.
+12
+
+2.3 Additional properties close to the origin
+Let V denote a potential function in Ck+1(Rdst, R) and let u0 be a point in Rdst. Let
+us recall (see subsection 1.3) that the unstable manifold W u, 0
+V
+(u0) of the equilibrium
+(u0, 0Rdst, 0) for the autonomous differential system (1.7)) is one-dimensional.
+As a
+consequence there exists a unique solution r �→ u(r) of the differential system (1.2) such
+that the image of the map r �→
+�u(r), ˙u(r), r) lies in the intersection W u, 0, +
+V
+(u0) of this
+unstable manifold with the half-space where r is positive (this intersection was defined in
+(1.10)); or, in other words, such that
+�u(r), ˙u(r)
+� goes to (u0, 0) as r goes to 0+. This
+solution is defined on some (maximal) interval (0, rmax), where rmax is either a finite
+quantity or +∞. The following lemma provides properties of this solution that will be
+used in the sequel. To ease its statement, let us assume that rmax is equal to +∞ (only
+this case will turn out to be relevant), and let us consider the continuous extension of
+u(·) to the interval [0, +∞) (and let us still denote by u(·) this continuous extension).
+Lemma 2.3. If u(·) is not identically equal to u0 (in other words, if u0 is not a critical
+point of V ), then there exists a positive quantity ronce such that, denoting by Ionce the
+interval [0, ronce), the following conclusions hold:
+1. the function ˙u(·) does not vanish on Ionce,
+2. and, for every r∗ in Ionce and r in [0, +∞),
+u(r) = u(r∗) =⇒ r = r∗.
+Proof. The linearized system (1.7) at the equilibrium (u0, 0Rdst, 0) reads:
+d
+dτ
+�
+�
+�
+δu
+δv
+δr
+�
+�
+� =
+�
+�
+�
+0
+0
+0
+0
+−(dsp − 1)
+∇V (u0)
+0
+0
+1
+�
+�
+�
+�
+�
+�
+δu
+δv
+δr
+�
+�
+� ,
+thus the tangent space at (u0, 0Rdst, 0) to W u, 0
+V
+(u0) (the unstable eigenspace of the matrix
+of this system) is spanned by the vector
+�0, ∇V (u0)/dsp, 1
+�; it follows that
+(2.6)
+˙u(r) = r
+dsp
+∇V (u0)
+�1 + or→0+(r)
+� .
+Thus, if r0 is a sufficiently small positive quantity, then ˙u(·) does not vanish on (0, r0]
+(so that conclusion 1 of Lemma 2.3 holds provided that ronce is not larger than r0), and
+the map
+(2.7)
+[0, r0] → Rdst ,
+r �→ u(r)
+is a C1-diffeomorphism onto its image. For r in [0, +∞), let us denote
+�u(r), ˙u(r)
+� by
+U(r). According to the decrease (2.2) of the Hamiltonian, there exists a quantity ronce in
+(0, r0) such that, for every r∗ in [0, ronce),
+(2.8)
+HV
+�U(r0)
+� < −V
+�u(r∗)
+� .
+13
+
+Take r∗ in [0, ronce] and r in [0, +∞), and let us assume that u(r) equals u(r∗). If r was
+larger than r0 then it would follow from the expression (2.1) of the Hamiltonian, its
+decrease (2.2), and inequality (2.8) that
+−V
+�u(r)
+� ≤ HV
+�U(r)
+� ≤ HV
+�U(r0)
+� < −V
+�u(r∗)
+� ,
+a contradiction with the equality of u(r) and u(r∗). Thus r is not larger than r0, and
+it follows from the one-to-one property of the function (2.7) that r must be equal to
+r∗; conclusion 2 of Lemma 2.3 thus holds, and Lemma 2.3 is proved.
+2.4 Additional properties close to infinity
+Let V1 denote a potential function in Ck+1(Rdst, R) and u1,∞ denote a nondegenerate
+minimum point of V1. According to the implicit function theorem, there exists a (small)
+neighbourhood νrobust(V1, u1,∞) of Vquad-R and a Ck-function V �→ u∞(V ) defined on
+νrobust(V1, u1,∞) and with values in Rdst such that u∞(V1) equals u1,∞ and, for every
+V in νrobust(V1, u1,∞), u∞(V ) is a local minimum point of V . The following proposi-
+tion is nothing but the local centre-stable manifold theorem applied to the equilibrium
+�u∞(V ), 0Rdst, 0
+� of the (autonomous) differential system (1.8), for V close to V1. Addi-
+tional comments and references concerning local stable/centre/unstable manifolds are
+provided in subsection 2.2 of [1].
+Proposition 2.4 (local centre-stable manifold at infinity). There exist a neighbourhood
+ν of V1 in Ck+1(Rdst, R), included in νrobust(V1, u1,∞), such that, if ε1 and c1 denote
+sufficiently small positive quantities, then, for every V in ν, there exists a Ck-map
+(2.9)
+wcs, ∞
+loc, V : BRdst(u1,∞, ε1) × [0, c1] → Rdst ,
+(u, c) �→ wcs, ∞
+loc, V (u, c) ,
+such that, for every (u0, v0, c0) in BRdst(u1,∞, ε1) × Rdst × [0, c1], the following two
+statements are equivalent:
+1. v = wcs, ∞
+loc, V (u, c);
+2. the solution r �→
+�u(r), v(r), c(r)
+� of the differential system (1.8) with initial condi-
+tion (u0, v0, c0) at time r0 = 1/c0 is defined up to +∞, remains in BRdst(u1,∞, ε1)×
+Rdst × [0, c1] of all r larger than r0, and goes to
+�u∞(V ), 0Rdst, 0
+� as r goes to +∞.
+In particular, wcs, ∞
+loc, V
+�u∞(V ), 0
+� is equal to 0Rdst. In addition, the map
+BRdst(u1,∞, ε1) × [0, c1] × ν → Rdst ,
+(u, c, V ) �→ wcs, ∞
+loc, V (u, c)
+is of class Ck (with respect to u and c and V ), and, for every V in ν, the graph of the
+differential at
+�u∞(V ), 0) of the map (u, c) �→ wcs, ∞
+loc, V (u, c) is equal to the centre-stable
+subspace of the linearization at
+�u∞(V ), 0Rdst, 0
+� of the differential system (1.8).
+14
+
+Let us denote by W cs, ∞, +
+loc, V, ε1, c1
+�u∞(V )
+� the graph of the map (2.9) (restricted to positive
+values of c), see figure 1.1; with symbols,
+(2.10)
+W cs, ∞, +
+loc, V, ε1, c1
+�u∞(V )
+� =
+��u, wcs, ∞
+loc, V (u, c), c
+� : (u, c) ∈ BRdst(u1,∞, ε1) × (0, c1]
+�
+.
+This set defines a local centre-manifold (restricted to positive values of c) for the equilib-
+rium
+�u∞(V ), 0Rdst, 0
+� of the differential system (1.8). Its uniqueness (for positive values
+of c) is ensured by the dynamics of the centre component c, which, according to the
+third equation of system (1.8), decreases to 0 (see figure 1.1). The global centre-stable
+manifold W cs, ∞, +
+V
+�u∞(V )
+� already defined in (1.13) can be redefined as the points of
+R2dst ×(0, +∞) that eventually reach the local centre manifold W cs, ∞, +
+loc, V, ε1, c1
+�u∞(V )
+� when
+they are transported by the flow of the differential system (1.8).
+Remark. If the state dimension dst is equal to 1, then a calculation shows that
+wcs, ∞
+loc, V (u, c) = −
+�u − u∞(V )
+� ��
+V ′′�u∞(V )
+� + dsp − 1
+2
+c + . . .
+�
+,
+where “. . . ” stands for higher order terms in u − u∞(V ) and c. In particular the quantity
+∂c∂uwcs, ∞
+loc, V
+�u∞(V ), 0
+� is equal to the (negative) quantity −(dsp − 1)/2. The display of
+the local centre-stable manifold at infinity on figure 1.1 fits with the sign of this quantity.
+3 Tools for genericity
+Let
+(3.1)
+Vfull = Ck+1(Rdst, R) ,
+and, for a positive quantity R, let
+(3.2)
+Vquad-R =
+�
+V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = u2
+2
+�
+.
+Let us recall the notation SV introduced in (1.4).
+Lemma 3.1. For every positive quantity R and for every potential V in Vquad-R, the
+following conclusions hold.
+1. The flow defined by the differential system (1.2) (governing radially symmetric
+stationary solutions of the parabolic system (1.1)) is global (that is, every solution
+is defined on (0, +∞)).
+2. For every u in SV , the following bound holds:
+(3.3)
+sup
+r∈(0,+∞)
+|u(r)| < R .
+15
+
+Proof. Let V be in Vquad-R. According to the definition (3.2) of Vquad-R, there exists a
+positive quantity K such that, for every u in Rdst,
+|∇V (u)| ≤ K + |u| .
+As a consequence, the following inequalities hold for the right-hand side of the first order
+differential system (1.5):
+����
+�
+v, −dsp − 1
+r
+v + ∇V (u)
+����� ≤ |v| + dsp − 1
+r
+|v| + K + |u| ≤ K +
+�
+2 + dsp − 1
+r
+�
+|(u, v)| ,
+and this bound prevents the solution from blowing up in finite time, which proves
+conclusion 1.
+Now, take a function u in SV . Let us still denote by u(·) the continuous extension of
+this solution to [0, +∞). For every r in [0, +∞), let
+q(r) = u(r)2
+2
+and
+Q(r) = rdsp−1 ˙q(r) .
+Then (omitting the dependency on r),
+˙q = u · ˙u
+and
+¨q = ˙u2 + u · ¨u = ˙u2 − dsp − 1
+r
+˙q + u · ∇V (u) ,
+so that
+˙Q = rdsp−1
+�
+¨q + dsp − 1
+r
+˙q
+�
+= rdsp−1� ˙u2 + u · ∇V (u)
+� .
+According to the definition (3.2) of Vquad-R, there exists a positive quantity δ (sufficiently
+small) so that, for every w in Rdst,
+(3.4)
+|w| ≥ R − δ =⇒ w · ∇V (w) ≥ w2
+2 .
+Let us proceed by contradiction and assume that supr∈(0,+∞) |u(r)| is not smaller than
+R. Since u(·) is stable at infinity and since the critical points of V belong to the open
+ball BRdst(0, R − δ), it follows that the set
+�r ∈ [0, +∞) : |u(r)| ≥ R
+�
+is nonempty; let rout denote the minimum of this set. For the same reason, the set
+�r ∈ (rout, +∞) : |u(r)| < R − δ
+�
+is also nonempty. Let rback denote the infimum of this last set. It follows from these
+definitions that rback is larger than rout and that, for every r in (rout, rback), according to
+inequality (3.4),
+(3.5)
+˙Q(r) ≥ rdsp−1
+�
+˙u2(r) + u2(r)
+2
+�
+> 0 .
+16
+
+If on the one hand rout equals 0 then |u(0)| is not smaller than R and, since Q(0) equals
+0, it follows from inequality (3.5) that Q(·) is positive on (0, rback), so that the same is
+true for ˙q(·). Thus q(·) is strictly increasing on [0, rback] and |u(rback)| must be larger
+than |u(rout)|, a contradiction with the definition of rback. If on the other hand rout is
+positive, then |u(rout)| is equal to R and ˙q(rout) is nonnegative so that the same is true
+for Q(rout), and it again follows from inequality (3.5) that Q(·) is positive on (0, rback),
+yielding the same contradiction. Conclusion 2 of Lemma 3.1 is proved.
+Notation. For every positive quantity R and every potential V in Vquad-R, let
+(3.6)
+SV : (0, +∞)2 × R2dst → R2dst ,
+�(rinit, r), (uinit, vinit)
+� �→ SV
+�(rinit, r), (uinit, vinit)
+�
+denote the (globally defined) flow of the (non-autonomous) differential system (1.5) for
+this potential V . In other words, for every rinit in (0, +∞) and (uinit, vinit) in R2dst, the
+function
+(0, +∞) → R2dst ,
+r �→ SV
+�(rinit, r1), (uinit, vinit)
+�
+is the solution of the differential system (1.5) for the initial condition (uinit, vinit) at r
+equals rinit. According to subsection 1.3, the flow SV may be extended to the larger set
+(0, +∞)2 × R2dst ∪ [0, +∞)2 × Rdst × {0Rdst} ;
+according to this extension, for every u0 in Rdst, the solution taking its values in the
+(one-dimensional) unstable manifold W u, 0, +
+V
+(u0) reads:
+(3.7)
+[0, +∞) → Rdst ,
+r �→ SV
+�(0, r), (u0, 0Rdst)
+� .
+4 Generic transversality among potentials that are quadratic
+past a given radius
+4.1 Notation and statement
+Let us recall the notation SV and SV, u∞ introduced in (1.4).
+Proposition 4.1. There exists a generic subset of Vquad-R such that, for every potential
+V in this subset, every radially symmetric stationary solution stable at infinity of the
+parabolic system (1.1) (in other words, every u in SV ) is transverse.
+4.2 Reduction to a local statement
+Let V1 denote a potential function in Vquad-R and u1,∞ denote a nondegenerate minimum
+point of V1. According to the implicit function theorem, there exists a (small) neighbour-
+hood νrobust(V1, u1,∞) of Vquad-R and a Ck-function u∞(·) defined on νrobust(V1, u1,∞) and
+with values in Rdst such that u∞(V1) equals u1,∞ and, for every V in νrobust(V1, u1,∞),
+u∞(V ) is a local minimum point of V . The following local generic transversality statement
+yields Proposition 4.1 (as shown below).
+17
+
+Proposition 4.2. There exists a neighbourhood νV1, u1,∞ of V1 in νrobust(V1, u1,∞) and
+a generic subset νV1, u1,∞, gen of νV1, u1,∞ such that, for every V in νV1, u1,∞, gen, every
+radially symmetric stationary solution stable close to u∞(V ) at infinity of the parabolic
+system (1.1) (in other words, every u in SV, u∞(V )) is transverse.
+Proof that Proposition 4.2 yields Proposition 4.1. Let us denote by Vquad-R-Morse the
+dense open subset of Vquad-R defined by the Morse property:
+(4.1)
+Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are nondegenerate} .
+Let V1 denote a potential function in Vquad-R-Morse. According to the Morse property
+its minimum points are isolated and since V1 is in Vquad-R they belong to the open ball
+BRd(0, R), so that those minimum points are in finite number. Assume that Proposi-
+tion 4.2 holds. With the notation of this proposition, let us consider the following two
+intersections, at each time over all minimum points u1,∞ of V1:
+(4.2)
+νV1 =
+�
+νV1, u1,∞
+and
+νV1, gen =
+�
+νV1, u1,∞, gen .
+Since those are finite intersections, νV1 is still a neighbourhood of V1 in Vquad-R and the
+set νV1, gen is still a generic subset of νV1. This shows that the set
+{V ∈ Vquad-R-Morse :
+every u in SV, u∞(V ) is transverse}
+is locally generic. Applying Lemma 4.3 of [1] as in Subsection 5.2 of this reference shows
+that this local genericity implies the global genericity stated in Proposition 4.1, which is
+therefore proved.
+4.3 Proof of the local statement (Proposition 4.2)
+4.3.1 Setting
+For the remaining part of this section, let us fix a potential function V1 in Vquad-R and a
+nondegenerate minimum point u1,∞ of V1. Let ν be a neighbourhood of V1 in Vquad-R,
+included in νrobust(V1, u1,∞), and let ε1 and c1 be positive quantities, with ν and ε1 and
+c1 small enough so that the conclusions of Proposition 2.4 hold. Let
+r1 = 1/c1
+and
+M = Rdst × BRdst(u1,∞, ε1)
+and
+Λ = ν ,
+and
+N = (R2dst)2
+and
+W = {(A, B) ∈ N : A = B}
+,
+thus W is the diagonal of N. Let N denote an integer not smaller than r1, and let us
+consider the functions
+Φu : Rdst × Λ → R2dst ,
+(u0, V ) �→ SV
+�(0, N), (u0, 0Rdst)
+� ,
+and
+Φcs : BRdst(u1,∞, ε1) × Λ → R2dst ,
+(uN, V ) �→
+�uN, wcs, ∞
+loc, V (uN, 1/N)
+� ,
+and the function
+(4.3)
+Φ : M × Λ → N ,
+(m, V ) = (u0, uN, V ) �→
+�Φu(u0, V ), Φcs(uN, V )
+� .
+18
+
+4.3.2 Equivalent characterizations of transversality
+Let us consider the set
+SΛ,u1,∞,N =
+�(V, u) : V ∈ Λ and u ∈ SV, u∞(V ) and u(N) ∈ BRdst(u1,∞, ε1)
+� .
+Proposition 4.3. The map
+(4.4)
+Φ−1(W) → SΛ,u1,∞,N ,
+(u0, u, V ) �→
+�
+V, r �→ SV
+�(0, r), (u0, 0Rdst
+��
+is well defined and one-to-one.
+Proof. The image by Φ of a point (u0, uN, V ) of M × Λ belongs to the diagonal W of
+N if and only if Φu(u0, V ) equals Φcs(uN, V ), and in this case the function u : r �→
+SV
+�(0, r), (u0, 0Rdst
+� belongs to SV, u∞(V ) and u(N) (which is equal to uN) belongs to
+BRdst(u1,∞, ε1), so that (V, u) belongs to SΛ,u1,∞,N. The map (4.4) above is thus well
+defined.
+Now, for every (V, u) in SΛ,u1,∞,N, if we denote by u0 the limit limr→0+ u(r) and by
+uN the vector u(N), then (u0, uN, V ) is the only possible antecedent of (V, u) by the map
+(4.4). In addition,
+SV
+�(0, N), (u0, 0Rdst)
+� =
+�uN, ˙u(N)
+� ,
+and since u(r) goes to u∞(V ) as r goes to +∞, the vector
+�u(N), ˙u(N), 1/N
+� must
+belong to the centre-stable manifold W cs, ∞, +
+V
+�u∞(V )
+� of u∞(V ), so that, according to
+the definition of wcs, ∞
+loc, V ,
+˙u(N) = wcs, ∞
+loc, V
+�u(N), 1/N
+� ,
+and this yields the equality between Φu(u0, V ) and Φcs(uN, V ). Thus Φ(V, u) belongs to
+W and (u0, uN, V ) belongs to Φ−1(W). Proposition 4.3 is proved.
+Proposition 4.4. For every potential function V in Λ, the following two statements are
+equivalent.
+1. The image of the function M → N, m �→ Φ(m, V ) is transverse to W.
+2. Every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse.
+Remark. According to Proposition 2.2, for every V in Λ, the constant function r �→ u∞(V ),
+which belongs to SV , is already (a priori) known to be transverse, therefore only
+nonconstant solutions matter in statement 2 of this proposition.
+Proof. Let us consider (m2, V2) in M × Λ such that Φ(m2, V2) is in W, let (u2,0, u2,N)
+denote the components of m2, and let r �→ u2(r) and r �→ U2(r) denote the functions
+satisfying, for all r in [0, +∞),
+U2(r) =
+�u2(r), ˙u2(r)
+� = SV
+�(0, r), (u2,0, 0Rdst
+� .
+Let us consider the map
+∆Φ : M → R2dst ,
+(u0, uN) �→ Φu(u0, V2) − Φcs(uN, V2) ,
+19
+
+and let us write, only for this proof, DΦ and DΦu and DΦcs and D(∆Φ) for the
+differentials of Φ and Φu and Φcs and ∆Φ at (m2, V2) and with respect to all variables in
+M (but not with respect to V ). According to Definition 1.5, the transversality of u2 is
+defined as the transversality of the intersection W u, 0, +
+V2
+∩ ι−1�
+W cs, ∞, +
+V2
+�u∞(V2)
+��
+along
+the trajectory of U2. This transversality can be considered at a single point, no matter
+which, of the trajectory U2
+�(0, +∞)
+�, in particular at the point Φu(u2,0, V2) which is
+equal to Φcs�u2(N), V 2
+�, and is equivalent to the transversality of the dst-dimensional
+manifolds
+W u, 0, +
+V2
+∩
+�R2dst × {N}
+�
+and
+ι−1�
+W cs, ∞, +
+V2
+�u∞(V2)
+��
+∩
+�R2dst × {N}
+�
+in R2dst ×{N}. It is therefore equivalent to the surjectivity of the map D(∆Φ) (statement
+(B) in Lemma 4.5 below). On the other hand, the image of the function M → N,
+m �→ Φ(m, V2) is transverse at Φ(m, V2) to the diagonal W of N if and only if the image
+of DΦ contains a complementary space of this diagonal (statement (A) in Lemma 4.5
+below)). Thus Proposition 4.4 is a consequence of the next lemma.
+Lemma 4.5. The following two statements are equivalent.
+(A) The image of DΦ contains a complementary subspace of the diagonal W of N.
+(B) The map D(∆Φ) is surjective.
+Proof. If statement (A) holds, then, for every (α, β) in N, there exist γ in R2dst and δm
+in Tm2M such that
+(4.5)
+(γ, γ) + DΦ · δm = (α, β) ,
+so that
+(4.6)
+D(∆Φ) · δm = α − β ,
+and statement (B) holds. Conversely, if statement (B) holds, then, for every (α, β) in
+N, there exists δm in Tm2M such that (4.6) holds, and as a consequence, if (δu0, δuN)
+denote the components of δm, then α − DΦu(δu0) is equal to β − DΦcs(δuN), and if
+this vector is denoted by γ, then equality (4.5) holds, and this shows that statement (A)
+holds.
+As explained above, Proposition 4.4 follows from Lemma 4.5, and is therefore proved.
+4.3.3 Checking hypothesis 1 of Theorem 4.2 of [1]
+The function Φ is as regular as the flow SV , thus of class Ck. It follows from the definitions
+of M and N and W that
+dim(M) − codim(W) = (dst + dst) − 2dst = 0 ,
+so that hypothesis 1 of Theorem 4.2 of [1] is fulfilled.
+20
+
+4.3.4 Checking hypothesis 2 of Theorem 4.2 of [1]
+For every V in Vquad-R, let us recall the notation SV introduced in (3.6) and (3.7) for the
+flow of the differential system (1.5). Take (m2, V2) in the set Φ−1(W). Let (u2,0, u2,N)
+denote the components of m2, and, for every r in (0, +∞), let us write
+U2(r) =
+�u2(r), v2(r)
+� = SV2
+�(0, r), (u2,0, 0Rdst)
+� .
+Let us write
+DΦ
+and
+DΦu
+and
+DΦcs
+for the full differentials (with respect to arguments m in M and V in Λ) of the three
+functions Φ and Φu and Φcs respectively at the points
+�u2,0, u2,N, V2
+�,
+�u2,0, V2
+� and
+�u2,N, V2
+�. Checking hypothesis 2 of Theorem 4.2 of [1] amounts to prove that
+(4.7)
+im(DΦ) + W = N .
+If u2(·) is constant (that is, identically equal to u∞(V2)), then equality (4.7) follows from
+Proposition 2.2. Thus, let us assume that u2(·) is nonconstant. In this case, equality
+(4.7) is a consequence of the following lemma.
+Lemma 4.6. For every nonzero vector (φ2, ψ2) in R2dst, there exists a function W in
+Ck+1
+b
+(Rdst, R) such that
+supp(W) ⊂ BRd(0, R) ,
+(4.8)
+and
+�DΦu · (0, 0, W)
+�� (φ2, ψ2)
+� ̸= 0 ,
+(4.9)
+and
+DΦcs · (0, 0, W) = 0R2dst .
+(4.10)
+Proof that Lemma 4.6 yields equality (4.7). Inequality (4.9) shows that the orthogonal
+complement, in R2dst, of the directions that can be reached by DΦu·(0, 0, W) for potentials
+W satisfying (4.8) and (4.10) is reduced to 0R2dst; in other words, all directions of R2dst
+can be reached by that means. This shows that
+im(DΦ) ⊃ R2dst × {0R2dst} ,
+and since the subspace at the right-hand side of this inclusion is transverse to W in
+R4dst, this proves equality (4.7) (and shows that hypothesis 2 of Theorem 4.2 of [1] is
+fulfilled).
+Proof of Lemma 4.6. Let (φ2, ψ2) denote a nonzero vector in R2dst, let W be a function
+in Ck+1
+b
+(Rdst, R) satisfying the inclusion
+(4.11)
+supp(W) ⊂ BRd(0, R) \ BRdst(u1,∞, ε1) ,
+and observe that inclusion (4.8) and equality (4.10) follow from this inclusion (4.11). Let
+us consider the linearization of the differential system (1.2), for the potential V2, around
+the solution r �→ U2(r):
+(4.12)
+d
+dr
+�
+δu(r)
+δv(r)
+�
+=
+�
+0
+id
+D2V2
+�u2(r)
+�
+−dsp−1
+r
+� �
+δu(r)
+δv(r)
+�
+,
+21
+
+and let T(r, r′) denote the family of evolution operators obtained by integrating this
+linearized differential system between r and r′. It follows from the variation of constants
+formula that
+(4.13)
+DΦu · (0, 0, W) =
+� N
+−∞
+T(r, N)
+�
+0, ∇W
+�u2(r)
+��
+dr .
+For every r in (0, +∞), let T ∗(r, N) denote the adjoint operator of T(r, N), and let
+(4.14)
+�φ(r), ψ(r)
+� = T ∗(r, N) · (φ2, ψ2) .
+According to expression (4.13), inequality (4.9) reads
+� N
+−∞
+��
+0, ∇W
+�u2(r)
+�� ��� T ∗(r, N) · (φ2, ψ2)
+�
+dr ̸= 0 ,
+or equivalently
+(4.15)
+� N
+−∞
+∇W
+�u2(r)
+� · ψ(r) dr ̸= 0 .
+Due to the expression of the linearized differential system (4.12), (φ, ψ) is a solution of
+the adjoint linearized system
+(4.16)
+� ˙φ(r)
+˙ψ(r)
+�
+= −
+�
+0
+D2V2
+�u2(r)
+�
+id
+−dsp−1
+r
+� �
+φ(r)
+ψ(r)
+�
+.
+According to Lemma 2.3 (and since u2(·) was assumed to be nonconstant), there exists
+positive quantity ronce such that, if we denote by Ionce the interval (0, ronce], then ˙u2(·)
+does not vanish on Ionce, and, for all r∗ in Ionce and r in R,
+(4.17)
+u2(r) = u2(r∗) =⇒ r = r∗ .
+In addition, up to replacing ronce by a smaller positive quantity, it may be assumed that
+the following conclusions hold:
+u2(Ionce) ∩ BRdst(u1,∞, ε1) = ∅ .
+To complete the proof three cases have to be considered.
+Case 1.
+There exists r∗ in Ionce such that ψ(r∗) is not collinear to ˙u2(r∗).
+In this case, the construction of a potential function W satisfying inclusion (4.11) and
+inequality (4.9) (and thus the conclusions of Lemma 4.6) is the same as in the proof of
+Lemma 5.7 of [1].
+If case 1 does not occur, then ψ(r) is collinear to ˙u2(r), and since ˙u2(·) does not vanish
+on Ionce, there exists a C1-function α : Ionce → R such that, for every r in Ionce,
+(4.18)
+ψ(r) = α(r) ˙u2(r) .
+The next cases 2 and 3 differ according to whether the function α(·) is constant or not.
+22
+
+Case 2.
+For every r in Ionce, equality (4.18) holds for some nonconstant function α(·).
+In this case there exists r∗ in Ionce such that ˙α(r∗) is nonzero, and again the construction
+of a potential function W satisfying inclusion (4.11) and inequality (4.9) (and thus the
+conclusions of Lemma 4.6) is the same as in the proof of Lemma 5.7 of [1].
+Case 3.
+For every r in Ionce, ψ(r) = α ˙u2(r) for some real (constant) quantity α.
+In this case the quantity α cannot be 0 or else, due to (4.16) and (4.18), both φ(·)
+and ψ(·) would identically vanish on Ionce and thus on (0, +∞), a contradiction with the
+assumptions of Lemma 4.6. Thus, without loss of generality, we may assume that α is
+equal to 1. If supp(W) is included in a sufficiently small neighbourhood of u2,0, then
+W(·) vanishes on u2
+�[ronce, N]
+� and the integral on the left-hand side of inequality (4.15)
+reads
+� ronce
+0
+∇W
+�u2(r)
+� · ˙u2(r) dr = W
+�u2(ronce)
+� − W(u2,0) = −W(u2,0) ,
+so that inequality (4.15) holds as soon as W(u2,0) is nonzero. Lemma 4.6 is proved.
+Remark. By contrast with the proof of the generic elementarity of standing pulses in
+[1], case 3 above cannot be easily precluded. Indeed, let us assume that, for every r in
+Ionce, ψ(r) is equal to α ˙u2(r) for some nonzero (constant) quantity α. Without loss of
+generality, we may assume that α is equal to 1. Then, it follows from the second equation
+of (4.16) that, still for every r in Ionce (omitting the dependency on r),
+φ = dsp − 1
+r
+ψ − ˙ψ = dsp − 1
+r
+˙u2 − ¨u2 = 2(dsp − 1)
+r
+˙u2 − ∇V2(u2) ,
+and it follows from the first equation of (4.16) that
+−D2V2(u2) ˙u2 = ˙φ = −2(dsp − 1)
+r2
+˙u2 + 2(dsp − 1)
+r
+¨u2 − D2V2(u2) ˙u2 ,
+and thus, after simplification,
+¨u2 = 1
+r ˙u2 ,
+or equivalently
+˙u2 = r
+dsp
+∇V (u2) .
+As illustrated by equality (2.6), this last equality indeed holds if ∇V2 is constant on the
+set u2(Ionce). Case 3 can therefore not be a priori precluded, and if it may be argued
+that this case is “unlikely” (non generic), the direct argument provided above in this
+case is simpler. By contrast, in [1] for standing pulses in space dimension one (dsp equal
+to 1), this case could not occur because ψ was assumed to be nonzero on the symmetry
+subspace, defined here as {(v, r) = (0Rdst, 0)}, see (1.11).
+4.3.5 Conclusion
+As seen in sub-subsection 4.3.3, hypothesis 1 of Theorem 4.2 of [1] is fulfilled for the
+function Φ defined in (4.3), and since Lemma 4.6 yields equality (4.7), hypothesis 2 of this
+23
+
+theorem is also fulfilled. The conclusion of this theorem ensures that there exists a generic
+subset Λgen, N of Λ such that, for every V in Λgen, N, the image of the function M → N,
+m �→ Φ(m, V ) is transverse to the diagonal W of N. According to Proposition 4.4, it
+follows that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) is transverse. The
+set
+Λgen =
+�
+N∈N, N≥r0
+Λgen, N
+is still a generic subset of Λ. For every V in Λgen and every u in SV, u∞(V ), since u(r)
+goes to u∞(V ) as r goes to +∞, there exists N such that u(N) is in BRdst(u1,∞, ε1), and
+according to the previous statements u is transverse. In other words, the conclusions of
+Proposition 4.2 hold with:
+νV1, u1,∞ = ν = Λ
+and
+νV1, u1,∞, gen = Λgen .
+5 Proof of the main results
+Proposition 4.1 shows the genericity of the property considered in Theorem 1, but only
+inside the space Vquad-R of the potentials that are quadratic past some radius R. In this
+section, the arguments will be adapted to obtain the genericity of the same property
+in the space Vfull (that is Ck+1(Rdst, R)) of all potentials, endowed with the extended
+topology (see subsection 1.5). They are identical to those of section 9 of [1]. Let us recall
+the notation SV introduced in (1.4), and, for every positive quantity R, let us consider
+the set
+SV,R =
+�
+u ∈ SV :
+sup
+r∈[0,+∞)
+|u(r)| ≤ R
+�
+.
+Exactly as shown in subsection 9.1 of [1], Theorem 1 follows from the next proposition.
+Proposition 5.1. For every positive quantity R, there exists a generic subset Vfull-⋔-S-R
+of Vfull such that, for every potential V in this subset, every radially symmetric stationary
+solution stable at infinity in SV,R is transverse.
+Proof. Let R denote a positive quantity, let V1 denote a potential function in Vquad-(R+1),
+and let u1,∞ denote a nondegenerate minimum point of V1. Let us consider the neigh-
+bourhood νV1, u1,∞ of V1 in Vquad-(R+1) provided by Proposition 4.2 for these objects,
+together with the quantities ε1, c1, and r1 introduced in sub-subsection 4.3.1. Up to
+replacing νV1, u1,∞ by its interior, we may assume that it is open in Vquad-(R+1). As in
+sub-subsection 4.3.1, let us consider an integer N not smaller than r1, and the same
+function Φ : M × Λ → N as in (4.3).
+Here is the sole difference with the setting of sub-subsection 4.3.1: by contrast with the
+non-compact set M defining the departure set of Φ, let us consider the compact subset
+MN defined as:
+MN = BRdst(0Rdst, N) × BRdst(u1,∞, ε1) .
+Thus the integer N now serves two purposes: the “time” (radius) at which the intersection
+between unstable and centre-stable manifolds is considered, and the radius of the ball
+24
+
+containing the departure points of the unstable manifolds that are considered. These
+purposes are independent (two different integers instead of the single integer N may as
+well be introduced). Let us consider the set:
+OV1,u1,∞,N =
+�
+V ∈ νV1, u1,∞ : Φ(MN, V ) is transverse to W in N
+�
+.
+As shown in Proposition 4.4, this set OV1,u1,∞,N is made of the potential functions V in
+νV1, u1,∞ such that every u in SV, u∞(V ) such that u(N) is in BRdst(u1,∞, ε1) and u(0) is in
+BRdst(0Rdst, N), is transverse. This set contains the generic subset νV1, u1,∞, gen = Λgen of
+νV1, u1,∞ and is therefore generic (thus, in particular, dense) in νV1, u1,∞. By comparison
+with νV1, u1,∞, gen, the additional feature of this set OV1,u1,∞,N is that it is open: exactly
+as in the proof of Lemma 9.2 of [1], this openness follows from the intrinsic openness of a
+transversality property and the compactness of MN.
+Let us make the additional assumption that the potential V1 is a Morse function. Then,
+the set of minimum points of V1 is finite and depends smoothly on V in a neighbourhood
+νrobust(V1) of V1. Intersecting the sets νV1, u1,∞ and OV1,u1,∞,N above over all the minimum
+points u1,∞ of V1 provides an open neighbourhood νV1 of V1 and an open dense subset
+OV1,N of νV1 such that, for all V in νV1, every radially symmetric stationary solution
+stable close to a minimum point of V at infinity, and equal at origin to some point of
+BRdst(0Rdst, N), is transverse.
+Denoting by int(A) the interior of a set A and using the notation of subsection 4.4 of
+[1], let us introduce the sets
+˜νV1 = res−1
+R,∞ ◦ resR,(R+1)(νV1) ,
+and
+˜OV1,N = res−1
+R,∞ ◦ resR,(R+1)(OV1,N) ,
+and
+˜Oext
+V1,N = ˜OV1,N ⊔ int
+�Vfull \ ˜νV1
+� .
+It follows from these definitions that ˜Oext
+V1,N is a dense open subset of Vfull (for more
+details, see Lemma 9.3 of [1]).
+Since Vquad-(R+1) is a separable space, it is second-countable, and can be covered by a
+countable number of sets of the form νV1. With symbols, there exists a countable family
+(V1,i)i∈N of potentials of Vquad-(R+1)-Morse so that
+Vquad-(R+1)-Morse =
+�
+i∈N
+νV1,i .
+Let us consider the set
+Vfull-⋔-S-R = Vfull-Morse ∩
+�
+�
+�
+(i,N)∈N2
+˜Oext
+V1,i,N
+�
+� ,
+where Vfull-Morse is the set of potentials in Vfull which are Morse functions. This set is a
+countable intersection of dense open subsets of Vfull, and is therefore a generic subset of
+Vfull. And, for every potential V in this set Vfull-⋔-S-R, every radially symmetric stationary
+solution stable at infinity in SV,R is transverse (for more details, see Lemma 9.4 of [1]).
+Proposition 5.1 is proved.
+25
+
+As already mentioned at the beginning of this section, Theorem 1 follows from Proposi-
+tion 5.1. Finally, Corollary 1.1 follows from Theorem 1 (for more details, see subsection 9.4
+of [1]).
+Acknowledgements
+This paper owes a lot to numerous fruitful discussions with Romain
+Joly, about both its content and the content of the companion paper [1] written in
+collaboration with him.
+References
+[1]
+R. Joly and E. Risler. “Generic transversality of travelling fronts, standing fronts,
+and standing pulses for parabolic gradient systems”. In: arXiv (2023), pp. 1–69.
+arXiv: 2301.02095 (cit. on pp. 3, 5, 7–10, 14, 18, 20–26).
+[2]
+E. Risler. “Global behaviour of bistable solutions for gradient systems in one
+unbounded spatial dimension”. In: arXiv (2022), pp. 1–91. arXiv: 1604.02002
+(cit. on p. 3).
+[3]
+E. Risler. “Global behaviour of bistable solutions for hyperbolic gradient systems
+in one unbounded spatial dimension”. In: arXiv (2022), pp. 1–75. arXiv: 1703.01221
+(cit. on p. 3).
+[4]
+E. Risler. “Global behaviour of radially symmetric solutions stable at infinity for
+gradient systems”. In: arXiv (2022), pp. 1–52. arXiv: 1703.02134 (cit. on p. 3).
+[5]
+E. Risler. “Global relaxation of bistable solutions for gradient systems in one
+unbounded spatial dimension”. In: arXiv (2022), pp. 1–69. arXiv: 1604.00804
+(cit. on p. 3).
+Emmanuel Risler
+Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan,
+F-69621 Villeurbanne, France.
+emmanuel.risler@insa-lyon.fr
+26
+

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+Low-energy quasi-circular electron correlations with charge order
+wavelength in Bi2Sr2CaCu2O8+δ
+K. Scott,1, 2 E. Kisiel,3 T. J. Boyle,1, 2, 4 R. Basak,3 G. Jargot,5 S. Das,3 S. Agrestini,6
+M. Garcia-Fernandez,6 J. Choi,6 J. Pelliciari,7 J. Li,7 Y. D. Chuang,8 R. D. Zhong,9
+J. A. Schneeloch,9 G. D. Gu,9 F. L´egar´e,5 A. F. Kemper,10 Ke-Jin Zhou,6 V. Bisogni,7
+S. Blanco-Canosa,11, 12 A. Frano,3, 13 F. Boschini,5, 14 and E. H. da Silva Neto1, 2, ∗
+1Department of Physics, Yale University, New Haven, Connecticut 06520, USA
+2Energy Sciences Institute, Yale University, West Haven, Connecticut 06516, USA
+3Department of Physics, University of California San Diego, La Jolla, California 92093, USA
+4Department of Physics and Astronomy, University of California, Davis, California 95616, USA
+5Centre ´Energie Mat´eriaux T´el´ecommunications,
+Institut National de la Recherche Scientifique, Varennes, Qu´ebec J3X 1S2, Canada
+6Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom
+7National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
+8Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
+9Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY, USA
+10Department of Physics, North Carolina State University, Raleigh, NC 27695, U.S.A.
+11Donostia International Physics Center, DIPC, 20018 Donostia-San Sebastian, Basque Country, Spain
+12IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain
+13Canadian Institute for Advanced Research, Toronto, ON, M5G 1M1, Canada
+14Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
+∗ Corresponding Author: eduardo.dasilvaneto@yale.edu
+arXiv:2301.08415v1  [cond-mat.str-el]  20 Jan 2023
+
+2
+ABSTRACT
+In the study of dynamic charge order correlations in the cuprates, most high energy-
+resolution resonant inelastic x-ray scattering (RIXS) measurements have focused on mo-
+menta along the high-symmetry directions of the copper oxide plane. However, electron
+scattering along other in-plane directions should not be neglected as they may contain in-
+formation relevant, for example, to the origin of charge order correlations or to our un-
+derstanding of the isotropic scattering responsible for strange metal behavior in cuprates.
+We report high-resolution resonant inelastic x-ray scattering (RIXS) experiments that re-
+veal the presence of dynamic electron correlations over the qx-qy scattering plane in under-
+doped Bi2Sr2CaCu2O8+δ with Tc = 54 K. We use the softening of the RIXS-measured bond
+stretching phonon line as a marker for the presence of charge-order-related dynamic electron
+correlations. The experiments show that these dynamic correlations exist at energies below
+approximately 70 meV and are centered around a quasi-circular manifold in the qx-qy scat-
+tering plane with radius equal to the magnitude of the charge order wave vector, qCO. We
+also demonstrate how this phonon-tracking procedure provides the necessary experimental
+precision to rule out fluctuations of short-range directional charge order (i.e. centered around
+[qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]) as the origin of the observed correlations.
+INTRODUCTION
+Dynamic fluctuations from periodic charge order (CO) pervade the phase diagram of
+cuprate superconductors, perhaps even more than superconductivity itself [1]. The detec-
+tion of these fluctuations over energy and momentum was enabled by several recent advances
+in the energy resolution of resonant inelastic x-ray scattering (RIXS) instruments operat-
+ing in the soft x-ray regime. In the case of YBa2Cu3O6+δ, Cu-L3 RIXS detects dynamic
+correlations at the charge order wavevector, qCO, with a characteristic energy scale of ap-
+proximately 20 meV [2]. It has been proposed that these low-energy short-range dynamic
+charge order correlations are a key ingredient to the strange metal behavior [3, 4] charac-
+terized by linear-in-temperature resistivity [5, 6]. On one hand, this temperature behav-
+ior is often associated with an isotropic scattering rate that depends only on temperature
+in units of energy and Planck’s constant (i.e. ∝ kBT/ℏ, sometimes called the Planckian
+regime) [7–11], as supported by recent angle-dependent magnetoresistance measurements of
+La1.6−xNd0.4SrxCuO4 [12]. On the other hand, combined transport and RIXS studies have
+
+3
+recently shown an unexpected link between linear-in-temperature resistivity and charge or-
+der in YBa2Cu3O6+δ [13, 14]. Combined, these latest results suggest that fluctuations of the
+charge order should somehow result in an effective isotropic scattering. Still, high-resolution
+RIXS experiments have largely focused on the fluctuations along the high-symmetry crystal-
+lographic directions only, leaving the full structure of electron correlations within the copper
+oxide plane unknown.
+Recently, in Bi2Sr2CaCu2O8+δ (Bi-2212), RIXS measurements found the existence of a
+quasi-circular pattern in the qx-qy plane at finite energies and with the same wave vector
+magnitude as that of the observed static charge order peak at q = [qx = ±qCO, qy =
+0] and [qx = 0, qy = ±qCO] – i.e.
+dynamic correlations with charge order wavelength
+along all direction in the CuO2 plane [15]. Although the medium energy-resolution of those
+measurements (∆E ≈ 0.8 eV) precluded a more precise determination of their energy profile,
+the results suggested that these quasi-circular dynamic correlations (QCDCs) appear broad
+over the mid-infrared ranges (defined approximately as 100 to 900 meV). This scattering
+manifold, which may result from combined short- and long-range Coulomb interactions [15–
+17], would provide a large variety of wave vectors for connecting all points of the Fermi
+surface (i.e. an effective isotropic scattering). However, it is not yet experimentally known
+if this manifold extends to electron scattering at lower energies, in the quasi-elastic regime.
+To experimentally investigate this scenario we used high energy-resolution (≈ 37 meV) Cu-L3
+RIXS qx-qy mapping of the electronic correlations in Bi-2212. Using the softening of the bond
+stretching (BS) phonon in RIXS as a marker of charge order correlations, our measurements
+reveal the presence of low-energy quasi-circular dynamic electronic correlations with |q| ≈
+qCO.
+RESULTS
+High-resolution RIXS mapping of dynamic correlations in the qx-qy plane
+We performed measurements at φ = 0◦, 25◦, 30◦, 35◦, 45◦, where φ is defined as the az-
+imuthal angle from the qx axis. For each φ, we acquired RIXS spectra at different values of
+in-plane momentum-transfer q = |q| by varying the incident angle on the sample. Through-
+out the paper, values of q are reported in reciprocal lattice units (r.l.u.), where one r.l.u. is
+defined as 2π/a and a = 3.82 ˚A (the lattice constant along φ = 0◦). In Fig. 1 (A and B), we
+
+4
+show representative spectra obtained at q near qCO for φ = 0◦ and 30◦, and energies below
+1.1 eV. In these two cases, the minimal model that fits the data includes five contributions:
+a quasi-elastic peak, a bond-stretching phonon peak at ≈ 70 meV, a peak at ≈ 135 meV
+(likely from a two-phonon process), a broad paramagnon and a broad background feature
+of unknown origin. A similar assessment can be made regarding all other high-resolution
+spectra acquired in this work. In this type of fitting analysis, the QCDCs are not explicitly
+accounted for and it is generally difficult to disentangle overlapping contributions to the
+RIXS spectra using a fitting model with so many parameters, thus precluding the extraction
+of the exact spectral profile of the QCDCs with any reasonable confidence. Still, we note
+that this high-resolution data is consistent with the previously reported medium-resolution
+data [15], which can be verified by integration of the high-resolution data (see supplementary
+materials, Fig. S7).
+It is likely that the spectral intensity of QCDCs in Bi-2212 is so dilute over energy
+as to preclude the extraction of their spectral structure amidst stronger paramagnon and
+phonon signals. Still, here we develop a different method to detect QCDCs at lower energies,
+by tracking the evolution of the bond-stretching (BS) phonon over the qx-qy plane. This
+method is based on the phenomenology revealed by several recent RIXS measurements of
+the cuprates along qx and qy, which indicate an apparent softening of the BS phonon peak
+at the momentum location of the static CO peak [18–23]. In the case of Bi-2212, it has
+been proposed that the apparent softening of the BS phonon in RIXS is due to an interplay
+between low-energy fluctuations of the charge order and BS phonons that results in a Fano-
+like interference [18, 21, 23, 24]. Another possibility is that the apparent softening is simply
+the result of the phonon peak and a low-energy charge order peak overlapping, as recently
+suggested by measurements of both YBa2Cu3O6+δ and Bi-2212 [25]. In either interpretation
+the location of the phonon softening can be used as a marker for low-energy charge order
+correlations.
+Figure 1 (C and D) shows the spectra acquired as a function of q for φ = 0◦ and 30◦,
+respectively, focusing on the region of the BS phonon.
+At φ = 0◦, it is clear that the
+phonon peak position softens to its lowest energy value at q = qCO ≈ 0.29 r.l.u. (Fig. 1C).
+Careful observation of the spectra taken along φ = 30◦ shows a similar softening effect with
+the lowest phonon energy position occuring for q ≈ qCO (Fig. 1D). Figure 2A shows the
+mapping of the BS phonon mode at φ = 0◦ and 30◦ obtained after subtraction of the fitted
+
+5
+3
+2
+1
+0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+ϕ=0°
+q=0.27 r.l.u.
+1
+0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+ϕ=30°
+q=0.27 r.l.u.
+IRIXS (arb. units)
+IRIXS (arb. units)
+Intensity (arb. units)
+Energy Loss (meV)
+A
+B
+C
+Energy Loss (eV)
+10
+8
+6
+4
+2
+120
+80
+40
+Energy Loss (eV)
+ϕ=0°
+D
+ϕ=30°
+120
+80
+40
+q (r.l.u.)
+0.1
+0.12
+0.14
+0.16
+0.17
+0.18
+0.19
+0.2
+0.21
+0.22
+0.23
+0.24
+0.25
+0.26
+0.27
+0.28
+0.29
+0.30
+0.31
+0.32
+0.33
+0.34
+0.35
+0.36
+0.37
+0.39
+0.41
+0.43
+0.45
+0.47
+FIG. 1. RIXS spectra and fitting. (A and B) Examples of spectra at q = 0.27 r.l.u. for φ = 0◦
+and 30◦, respectively (open circles). The red lines are fits to the spectra, composed of a quasi-
+elastic peak (pink), a BS phonon peak at ≈ 70 meV (blue), a peak at ≈ 135 meV (likely from
+a two-phonon process) (purple), a broad paramagnon (orange) and a broad background feature
+of unknown origin (brown). (C and D) RIXS measured BS phonon peak for various values of q
+measured for φ = 0◦ and 30◦, respectively (black circles). The blue lines are the fits to the spectra.
+The vertical orange dashed lines, indicating the lowest phonon peak position at each φ, are shown
+to help the reader observe the phonon dispersions in the raw data.
+elastic line, once again showing the softening of the RIXS phonon even at φ = 30◦. To
+precisely determine the locations of the softening in the qx-qy plane, we fit the spectra to
+extract the dispersion of the BS phonon for each φ (Fig. 2B). We observe a softening of the
+RIXS measured phonon line for all φ, except for φ = 45◦. Remarkably all observed softening
+occurs at a value of q ≈ qCO, precisely as expected for QCDCs at low energies.
+Discriminating QCDCs from short-range directional order
+Dynamic correlations emanating from short range order are bound to be broad in q. It
+is therefore reasonable to ask whether the measured qx-qy profile of the BS phonon could
+
+6
+0.4
+0.3
+0.2
+0.1
+100
+60
+20
+100
+60
+20
+70
+60
+50
+0.4
+0.3
+0.2
+0.1
+ϕ:
+q (r.l.u.)
+q (r.l.u.)
+Energy Loss (meV)
+Phonon Energy (meV)
+1
+2
+1
+2
+3
+A
+B
+ϕ=0°
+ 0°
+ 25°
+ 30°
+ 35°
+ 45°
+ϕ=30°
+FIG. 2. Location of low-energy dynamic correlations extracted from the phonon dis-
+persion. (A) Energy-momentum structure of the excitations at φ = 0◦ and 30◦ after subtraction
+of the elastic line. The image is constructed from RIXS spectra deconvoluted from the energy res-
+olution. (B) Location of the phonon peak obtained by fitting the RIXS spectra deconvoluted from
+energy resolution for different φ (see Materials and Methods and also Supplementary Materials,
+Fig. S3). The solid lines are obtained by fitting the q-dependence of the phonon peak (circles) with
+a negative Lorentzian function plus a linear background. The shaded regions around the solid lines
+are generated from the 95% confidence interval obtained for the various fits to the spectra (see
+Materials and Methods for details). The solid lines for φ = 0◦ and 30◦ in (B) appear as dashed
+white lines in (A).
+simply be the result of diffuse scattering from short-range directional order. The fundamental
+difference between QCDCs and short-range directional order is that the former forms a
+manifold of dynamic correlations centered at q = qCO (similar to Brazovskii-type fluctuations
+[15, 26]), while the the latter results in dynamic correlations around q = [qx = ±qCO, qy = 0]
+and q = [qx = 0, qy = ±qCO] (more details on M1 and M2 are provided in the Materials and
+Methods section). To contrast these scenarios we consider two simple toy models. In both
+cases we start with a flat |q|-independent phonon mode at 72 meV, which is a reasonable
+approximation given the small dispersion of the BS phonon in the absence of charge order
+[25, 27]. In the first model (M1) we construct the QCDCs scenario, where the q-cuts for
+
+7
+-0.5
+-0.25
+0
+0.25
+0.5
+-0.5
+-0.25
+0
+0.25
+0.5
+qx (r.l.u.)
+qy (r.l.u.)
+-0.5
+-0.25
+0
+0.25
+0.5
+40
+45
+50
+55
+60
+65
+70
+Energy (meV)
+qx (r.l.u.)
+0.1
+0.2
+0.3
+0.4
+0.5
+40
+45
+50
+55
+60
+65
+70
+75
+Energy (meV)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.1
+0.25
+0.4
+0°
+90°
+180°
+270°
+φ
+q (r.l.u.)
+A
+B
+C
+D
+E
+T=Tc
+T<Tc
+M1
+M2
+M1
+M2
+qx (r.l.u.)
+qx (r.l.u.)
+45°
+40°
+35°
+30°
+25°
+15°
+0°
+φ:
+FIG. 3. Models of phonon softening for QCDCs and directional order. (A and C) Phonon
+dispersion for M1 and M2, as described in the text. (B and D) Momentum q cuts of the phonon
+dispersion at different φ for the simulated data in (A) and (C) respectively. The dashed orange
+and green lines in (A-D) identify the location of the phonon softening in the qx-qy plane. (E) Polar
+plot contrasting M1, M2 models (orange and green solid lines) and the experimental data (red
+symbols). The error bars in (E) are obtained from the fits to the phonon dispersion in Fig. 2B. See
+Materials and Methods for more details.
+various φ always have a minimum located at q = qCO, Fig. 3B. In the second model (M2) we
+consider the case where dynamic charge order correlations emerge isotropically from static
+peaks at [qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO]. The corresponding phonon profile is
+shown in Fig. 3 (C and D). To roughly emulate the data we also introduce a φ-dependent
+phonon minimum in M1, which increases from φ = 0◦ to 45◦, Fig. 3A. However, note that the
+magnitude of the softening depends on the φ structure of the electron-phonon coupling, which
+is not known or necessary for discerning the two scenarios. The q-cuts show a qualitatively
+similar behavior in both models: a clear phonon softening at φ = 0 that continues to
+exist even as φ approaches 45◦. However, in M2 the q location of the phonon minima clearly
+decreases with increasing φ from 0◦ to 45◦. This comparison explains our selection of φ values
+for these studies: the experimental ability to differentiate between M1 and M2 is largest in
+the φ = 25◦ to 45◦ range. The polar plot in Fig. 3E summarizes the analysis, comparing the
+
+8
+q-location of the minima for both models to the minima obtained from experiments (red
+markers). Within the error bars, the RIXS measurements are consistent with M1 and rule
+out M2, indicating the quasi-circular nature of the low-energy correlations associated with
+the charge order.
+DISCUSSION
+The experiments presented here provide evidence for the existence of quasi-circular dy-
+namic correlations at low energies in underdoped Bi-2212, which could be a key ingredient
+for models that connect charge order to an effective isotropic scattering. Long-range trans-
+lational symmetry breaking cannot be responsible for this isotropy due to the characteristic
+length scale and directionality of the ordered state. Although short-range electron correla-
+tions from directional order, occupying a much larger region of momentum space, could in
+principle emulate isotropic scattering [3, 4, 28], the QCDCs revealed by our experiments of-
+fer a different scenario. Extending not only around the static charge order wave vectors but
+also in the azimuthal direction, QCDCs might be a more viable platform for isotropic scat-
+tering. To fully understand the impact of QCDCs to electronic properties of the cuprates,
+one requires knowledge of the energy structure of these correlations. Although this might
+still be beyond the current experimental capabilities, our experiments provide some con-
+straints to the low-energy structure of the QCDCs. In particular, for the φ values where
+a softening is detected, QCDCs must exist below ≈ 70 meV (i.e. the approximate energy
+of the bare phonon). Unfortunately the amount of energy softening at q = qCO by itself,
+without knowledge of the φ-dependence of the electron-phonon interaction, does not provide
+more information about the energy structure of the QCDCs. Therefore, it remains possible
+that QCDCs at φ = 45◦ exist below 70 meV but do not significantly interact with the BS
+phonon.
+The quasi-circular shape of the low-energy correlations is similar to the shape obtained
+from the analysis of higher energy correlations (Ref. [15] and Supplementary Materials,
+Fig. S6). This similarity raises the possibility that the QCDCs exist up to much higher
+energies of order of 1 eV. As discussed in Ref. [15], the quasi-circular correlations cannot be
+explained by an instability of the Fermi surface. Instead, it was proposed that the loca-
+tion of the dynamic CO correlations in q-space is determined by the minima of the effective
+Coulomb interaction, which becomes non-monotonic in q due to the inclusion of a long-range
+
+9
+Coulomb interaction. However, this non-monotonic Coulomb interaction by itself failed to
+capture the intensity anisotropy observed at q ≈ qCO. Likewise, here the same proposed
+Coulomb interaction could also explain the most salient feature of our data, namely the
+quasi-circular shape of the low-energy correlations. Recently, a more complete theoretical
+description based on a t-J model with long-range Coulomb interaction shows the presence
+of ring-like charge correlations with the correct intensity anisotropy [17]. The results pre-
+sented here can serve as a guide for future theoretical investigations that also account for
+the apparent decrease of the phonon softening from φ = 0◦ to 45◦.
+Beyond the fact that both the energy-integrated correlations [15] and the low-energy
+dynamic correlations appear to occupy the same quasi-circular scattering manifold, the cur-
+rent RIXS measurements do not provide further experimental evidence to connect these
+two phenomena. Such additional evidence may come from polarimetric RIXS experiments
+that are able to decompose charge and spin excitations in the mid-infrared range, as it
+has been done for electron-doped cuprates [29]. Compared to the energy-integration pro-
+cedure, the phonon tracking method provides larger precision for mapping CO correlations
+in the qx-qy plane, since the large integration ranges required for the former result in very
+broad features in q-space.
+Indeed, we have already performed medium-resolution RIXS
+measurements that detect the presence of similar quasi-circular scattering manifolds in the
+energy-integrated spectrum of optimally and overdoped samples, but the investigation of
+their doping dependence is hindered by the large experimental uncertainty associated with
+the integration method (see Supplementary Materials, Fig. S6). Instead, our new procedure
+to track QCDCs using measurements of the RIXS BS phonon goes beyond demonstrating
+the existence of QCDCs in underdoped Bi-2212 at low energies. It is also a new methodology
+that can be used to detect QCDCs in other cuprates and understand related phenomena
+such as the electron-doped cuprates which also show a quasi-circular scattering [30]. Finally,
+the application of this new method to multiple cuprate families at different dopings and/or
+temperatures will help unveil whether and how QCDCs and the strange metal are related.
+MATERIALS AND METHODS
+RIXS experiments
+High-resolution RIXS experiments were performed at the I21 beamline [31] at Diamond
+
+10
+Light Source, United Kingdom, and at the 2-ID beamline at the National Synchrotron Light
+Source II, Brookhaven National Laboratory, USA. The orientation of the crystal axes of
+the underdoped Bi2Sr2CaCu2O8+δ samples with Tc = 54 K was obtained by x-ray diffraction
+prior to the RIXS experiment. The samples were cleaved in air just moments before inserting
+them into the ultra-high-vacuum chambers. For experiments at I21, the crystal was aligned
+to the scattering geometry in situ from measurements of the 002 Bragg reflection and the
+b-axis superstructure peak. The scattering angle was fixed at 154◦ (I21) and 153◦ (2-ID).
+The incoming light was set to vertical polarization (σ geometry) at the Cu-L3 edge (≈
+931.5 eV). The combined energy resolution (FWHM) was about 37 meV (I21) and 40 meV
+(2-ID), with small variations (±3 meV) over the course of multiple days. In both cases the
+energy resolution was relaxed in a trade-off for intensity. The projection of the momentum
+transfer, q, qx-qy plane was obtained by varying the incident angle on the sample (θ). All
+the measurements were performed at T = 54 K, which is the superconducting transition
+temperature for this sample, except for one measurement performed at T = 25 K below Tc
+(Fig. 3)E and one measurement at 300 K (Supplementary Materials, Figs. S2 and S5).
+Analysis of RIXS spectra
+To ensure the robustness of the extraction of phonon dispersion from the RIXS spectra we
+analyzed the data using multiple methods. Although the overall RIXS cross-section may
+depend on φ, we did not perform any normalization or intensity correction procedure to the
+spectra since the energy location of the phonon does not depend on the overall intensity.
+A comparison between the results for different methods is available in the supplementary
+materials, Fig. S4.
+Method 1: In an effort to maintain an agnostic approach and to not assume particular
+functional forms of the different contributions to the RIXS spectra, we extracted the dis-
+persion by simply tracking the energy positions of the phonon peak maximum in the RIXS
+spectra deconvoluted from the energy resolution. See below for details of the deconvolution
+procedure.
+Method 2: The phonon dispersion shown in Fig. 2B was extracted by fitting the deconvo-
+luted RIXS spectra (see Fig. 2A and supplementary materials, Fig. S3) in the [-30,130] meV
+range to a double Gaussian function plus a second order polynomial background, keeping
+all parameters free. The shaded regions around the solid lines in Fig. 2B were generated by
+fitting the 95% confidence intervals (obtained from the fits) to a polynomial function of q.
+
+11
+Method 3: Following previous works [23, 32], the raw RIXS spectra were fit to a five
+component model that includes a Gaussian (elastic peak of amplitude Ael, position ωel and
+width wel), two anti-Lorentzians (phonon and double-phonon peaks of different amplitude
+Ai and position ωi, and sharing width wph and Fano parameter width qF – note that i=1,2
+indicates the first and second phonon, respectively), a damped harmonic oscillator lineshape
+(paramagnon of amplitude Apm, position ωpm and damping parameter γpm), and an error
+function (smooth background described by an error function with amplitude amplitude ABG,
+position ωBG and width wBG):
+f(ω) = Aele
+− (ω−ωel)2
+w2
+el
++
+�
+i=1,2
+Ai
+2(ω − ωi)/wph + qF
+[2(ω − ωi)/wph]2 + 1+
++Apm
+γpmω
+(ω2 − ω2
+pm)2 + 4γ2
+pmω2 + ABG
+�
+erf
+�ω − ωBG
+wBG
+�
++ 1
+�
+(1)
+The fitting model is convoluted with the RIXS energy resolution (∼37 meV). From this
+analysis we extracted the phonon dispersion for each φ that quantitatively matches the
+phonon dispersion shown in Fig. 2B. All parameters are kept free, except for ωbg, which is
+constrained within a range of [0.2, 0.6] eV.
+Fitting the phonon dispersion
+To obtain a phenomenological form to the dispersions in Fig. 2B, the extracted peak locations
+as a function of q were fit to a linear background plus a negative Lorentzian function. From
+this fit we obtain the q location of the softening (red markers in Fig. 3E). To obtain the
+error bars in Fig. 3E, we follow a conservative approach by taking the average of the two
+q-intercepts of the fitted curve at Emin + 2 meV, where Emin is the lowest energy of the
+dispersion and ±2 meV is the typical amount of scatter observed in the data. For φ = 45◦
+the data is fit to a line.
+Deconvolution procedure
+We employed the Lucy-Richardson deconvolution procedure [33] to deconvolve the energy
+resolution (∼37 meV) from the RIXS spectra (deconvoluted curves for all azimuths are
+displayed in the supplementary materials, Fig. S3). The number of iterations and region
+of interest of the deconvolution procedure were optimized by ensuring that the convolution
+of the deconvoluted curves reproduced the raw data.
+Model simulations
+The toy models M1 and M2 are purely phenomenological. For M1, the phonon dispersion
+
+12
+was modeled as:
+E = E0 − ξ(φ)
+∆
+( q−q0
+Γ )2 + 1
+(2)
+where E0 = 72 meV, ∆ = 30 meV, Γ = 0.065 r.l.u., q0 = 0.29 r.l.u. and ξ(φ) = (| cos(2φ)| +
+0.08)/(1.08). For M2, the phonon dispersion was modeled as:
+E = E0 −
+4
+�
+i=1
+∆
+( q−qi
+Γ )2 + 1
+(3)
+where E0 = 74 meV, ∆ = 30 meV, Γ = 0.065 r.l.u. and qi are the four peaks located at
+[qx = ±qCO, qy = 0] and [qx = 0, qy = ±qCO] (qCO = 0.29 r.l.u.)
+Acknowledgments
+We acknowledge the Diamond Light Source for time on beamline I21-RIXS under propos-
+als MM28523 and MM30146. This research used resources of the Advanced Light Source,
+a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231. This re-
+search used beamline 2-ID of the National Synchrotron Light Source II, a U.S. Department
+of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by
+Brookhaven National Laboratory under Contract No. DE-SC0012704. We especially ac-
+knowledge the incredible work done by the beamline staffs at I-21, 2-ID and at 8.0.1 qRIXS,
+to allow many of these experiments to be performed remotely during the COVID pandemic.
+This work was supported by the Alfred P. Sloan Fellowship (E.H.d.S.N.). E.H.d.S.N ac-
+knowledges support by the National Science Foundation under Grant No. 2034345. A.F.K.
+was supported by the National Science Foundation under grant no. DMR-1752713. F.B.
+acknowledges support from the Fonds de recherche du Qu´ebec – Nature et technologies
+(FRQNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC).
+A.F. was supported by the Research Corporation for Science Advancement via the Cottrell
+Scholar Award (27551) and the CIFAR Azrieli Global Scholars program. This material is
+based upon work supported by the National Science Foundation under Grant No. DMR-
+2145080. The synthesis work at Brookhaven National Laboratory was supported by the US
+Department of Energy, office of Basic Energy Sciences, contract no. DOE-SC0012704.
+
+13
+Supplementary Materials
+12
+10
+8
+6
+4
+2
+0
+0.8
+0.6
+0.4
+0.2
+0.0
+10
+8
+6
+4
+2
+0
+0.8
+0.6
+0.4
+0.2
+0.0
+Intensity (arb. units)
+Energy Loss (eV)
+ϕ=0°
+ϕ=30°
+FIG. S1. Fits to the spectra in Fig. 2 The two panels show the fit of the RIXS spectra over a
+wide energy range using Method (3) as detailed in the Materials and Methods Section of the main
+text.
+
+14
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+120
+80
+40
+1.6
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+Intensity (arb. units)
+Energy Loss (eV)
+120
+80
+40
+ϕ=0°
+ϕ=30°
+70
+60
+50
+40
+0.4
+0.3
+0.2
+ 54 K
+ 300 K
+80
+q (r.l.u.)
+Phonon Dispersion (meV)
+ϕ=30°
+ϕ=0°
+ϕ=45°
+A
+B
+FIG. S2. Data obtained at the 2-ID beamline at NSLS-II. (A) RIXS spectra measured as a
+function of q for different φ at 54 K. The dashed line is a guide to the eye highlighting the phonon
+softening, visible at both φ = 0◦ and φ = 30◦ already from the raw data without need of doing any
+deconvolution. (B) Phonon dispersion obtained by using Method (2) as detailed in the Materials
+and Methods Section of the main text.
+
+15
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+5
+4
+3
+2
+1
+0
+0.12
+0.08
+0.04
+0.00
+16
+14
+12
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+10
+8
+6
+4
+2
+0
+0.12
+0.08
+0.04
+0.00
+Intensity (arb. units)
+Intensity (arb. units)
+Energy Loss (eV)
+ϕ=0°
+ϕ=45°
+ϕ=35°
+ϕ=30°
+ϕ=25°
+FIG. S3.
+Waterfall plot of the RIXS spectra deconvoluted for energy resolution
+(37 meV) for different φ.
+The softening of the BS phonon is clearly visible for any φ (ex-
+cept φ=45o) by simple visual inspection of the deconvoluted curves (red dash lines are guides to
+the eye).
+
+16
+ϕ:
+q (r.l.u.)
+Phonon Dispersion (meV)
+75
+70
+65
+60
+55
+50
+45
+40
+0.4
+0.3
+0.2
+0.1
+ 0°
+ 25°
+ 30°
+ 35°
+ 45°
+ 
+Fits:
+(I): 
+ 
+(II): 
+(III):
+70
+60
+50
+40
+75
+70
+65
+60
+55
+76
+72
+68
+64
+60
+76
+72
+68
+64
+76
+72
+68
+64
+0.4
+0.3
+0.2
+0.1
+ϕ = 0°
+ϕ = 25°
+ϕ = 30°
+ϕ = 35°
+ϕ = 45°
+q (r.l.u.)
+Phonon Dispersion (meV)
+A
+B
+FIG. S4. Comparison between three different methods for the extraction of the phonon
+dispersion. Methods (1), (2) and (3) are detailed in the Materials and Methods section of the
+main text.
+
+17
+0.25
+0.4
+0°
+90°
+180°
+270°
+φ
+q (r.l.u.)
+0.1
+I21 - T=Tc
+I21 - T<Tc
+M1
+M2
+2-ID - T=Tc
+2-ID - T>Tc
+FIG. S5. Comparison of models to experiments at 2-ID and I21. Polar plot contrasting
+M1, M2 models (orange and green solid lines) and the experimental data (blue and red symbols).
+The error bars are obtained from the fits to the phonon dispersion as described in the Materials and
+Methods section. On the left side, the model was adjusted for a higher value of qCO for comparison
+with the data obtained at 2-ID. The data at 2-ID is consistent with M1 and not with M2. In the
+main text only the data from I21 is shown because for those experiments the sample crystal axes
+could be aligned in situ from structural diffraction peaks by using the photodiode detector in that
+chamber. See Materials and Methods section for details.
+
+18
+Medium resolution RIXS
+In Fig. S6 we show the results of medium resolution RIXS done at the qRIXS endstation
+at the Advanced Light Source in the Lawrence Berkeley National Laboratory. The data
+were obtained by integrating the RIXS spectra over the −0.5 to 0.7 eV energy window and
+normalizing them by spectra integrated over all energies, which allows a comparison between
+the three different dopings. The data was also symmetrized about the high-symmetry φ =
+45◦ direction. In Fig. S6(D-I) the solid lines are fits of a Gaussian function plus a linear
+background to the data. The maps in Fig. S6(A-C) were generated from the fits in Fig. S6(G-
+I), respectively.
+The gray bars in Fig. S6(D-F) are centered at the average radii of the
+correlations, obtained from averaging over φ the peak positions obtained from the fits in
+Fig. S6(G-I). The widths of the grey bars in Fig. S6(D and E) are obtained from the 95%
+confidence intervals obtained from the fits in Fig. S6(G and H), summing them in quadrature
+and taking their square root. The same procedure underestimates the uncertainty for the
+Tc = 54 K. Instead the width of the grey bar in Fig. S6(F) is calculated by taking the lowest
+and largest peak positions over all φ, taking into account the 95% confidence intervals from
+the fits in Fig. S6(I). The width of the grey bar in Fig. S6(F) The data used to generate
+Fig. S6(C, F and I) were used in a previous publication [Boschini et al. Nat. Comm. 12, 1-
+8 2021]. The new data follows the same experimental procedures as the previously published
+data, so we direct the reader to [Boschini et al. Nat. Comm. 12, 1-8 2021] for further details
+of the experimental procedure.
+
+19
+0
+0.2 0.4
+q (r.l.u.)
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+0.1 0.2 0.3 0.4
+q (r.l.u.)
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+Intensity (a.u.)
+0
+0.2 0.4
+q (r.l.u.)
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+0.1 0.2 0.3 0.4
+q (r.l.u.)
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+Intensity (a.u.)
+0
+0.2 0.4
+q (r.l.u.)
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+0.1 0.2 0.3 0.4
+q (r.l.u.)
+18
+20
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+Intensity (a.u.)
+100°
+45°
+-10°
+0.1
+0.2
+0.3
+0.4
+22
+23
+24
+25
+26
+27
+q (r.l.u.)
+Intensity (a.u.)
+100°
+45°
+-10°
+0.1
+0.2
+0.3
+0.4
+22
+23
+24
+25
+26
+27
+q (r.l.u.)
+Intensity (a.u.)
+95°
+45°
+-10°
+0.1
+0.2
+0.3
+0.4
+22
+23
+24
+25
+26
+27
+q (r.l.u.)
+Intensity (a.u.)
+[-10°, 5°]
+[15°, 25°]
+[35°, 55°]
+[-10°, 10°]
+[20°, 30°]
+[40°, 50°]
+[-10°, 10°]
+[15°, 30°]
+[45°]
+-10°
+0°
+5°
+15°
+25°
+35°
+45°
+55°
+65°
+75°
+85°
+90°
+95°
+-10°
+0°
+10°
+20°
+30°
+40°
+50°
+60°
+70°
+80°
+90°
+100°
+-10°
+-5°
+0°
+5°
+10°
+15°
+30°
+45°
+60°
+75°
+80°
+90°
+95°
+85°
+100°
+Overdoped
+Tc = 60K
+Optimally doped
+Tc = 91K
+Underdoped
+Tc = 54K
+A
+B
+C
+D
+E
+F
+G
+H
+I
+FIG. S6. Doping dependence from medium resolution RIXS (A-C) Normalized energy-
+integrated RIXS mapping showing high energy quasi-circular electron correlations in overdoped,
+optimally doped and underdoped samples, respectively. (D-F) q-cuts integrated over different φ
+ranges, as specified in the legends. (G-I) The normalized energy-integrated RIXS data used to
+used to generate (A and D), (B and E) and (C and F).
+
+20
+1.0
+0.8
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+1.0
+0.5
+0.0
+0.8
+0.4
+0.0
+q (r.l.u.)
+Intensity (arb. units)
+∫100 meVIRIXS / ∫IRIXS
+700 meV
+Intensity (arb. units)
+ ϕ=0°
+ 25°
+ 30°
+ 35°
+ 45°
+q (r.l.u.)
+ϕ = 30°
+A
+B
+FIG. S7. Energy-integrated RIXS maps from high resolution RIXS on Bi2212 under-
+doped Tc=54 K at I21. A Energy-Loss RIXS spectrum for (q,φ)=(0.28 rlu, 30o). The orange
+shadow highlights the energy integration window [0.1,0.7] eV. B q-cuts of the RIXS spectra in-
+tegrated over the energy regions highlighted in A and normalized by the total energy-integrated
+RIXS signal. The overall q-dependence and position of the maximum is similar to what observed via
+medium resolution RIXS (see Fig. S6(F and I)). The pink bar reproduces the grey bar in Fig. S6(F),
+which is obtained from the analysis of the correlations observed with medium resolution RIXS for
+underdoped Bi2212 (Tc=54 K).
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+

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+Look, Listen, and Attack: Backdoor Attacks Against Video Action Recognition
+Hasan Abed Al Kader Hammoud1
+Shuming Liu1
+Mohammad Alkhrasi2
+Fahad AlBalawi2
+Bernard Ghanem1
+1 King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
+2 Saudi Data and Artificial Intelligence Authority (SDAIA), Riyadh, Saudi Arabia
+{hasanabedalkader.hammoud,shuming.liu,bernard.ghanem} @kaust.edu.sa
+{mkhrashi,falbalawi} @sdaia.gov.sa
+Abstract
+Deep neural networks (DNNs) are vulnerable to a class
+of attacks called “backdoor attacks”, which create an as-
+sociation between a backdoor trigger and a target label the
+attacker is interested in exploiting. A backdoored DNN per-
+forms well on clean test images, yet persistently predicts an
+attacker-defined label for any sample in the presence of the
+backdoor trigger. Although backdoor attacks have been ex-
+tensively studied in the image domain, there are very few
+works that explore such attacks in the video domain, and
+they tend to conclude that image backdoor attacks are less
+effective in the video domain. In this work, we revisit the
+traditional backdoor threat model and incorporate addi-
+tional video-related aspects to that model. We show that
+poisoned-label image backdoor attacks could be extended
+temporally in two ways, statically and dynamically, leading
+to highly effective attacks in the video domain. In addition,
+we explore natural video backdoors to highlight the seri-
+ousness of this vulnerability in the video domain. And, for
+the first time, we study multi-modal (audiovisual) backdoor
+attacks against video action recognition models, where we
+show that attacking a single modality is enough for achiev-
+ing a high attack success rate.
+1. Introduction
+A fundamental requirement for the deployment of deep
+neural networks (DNNs) in real-world tasks is their safety
+and robustness against possible vulnerabilities and security
+breaches. This requirement is, in essence, the motivation
+behind exploring adversarial attacks. One particularly in-
+teresting adversarial attack is “backdoor attacks”. Backdoor
+attacks or neural trojan attacks explore the scenario in which
+a user with limited computational capabilities downloads
+pretrained DNNs from an untrusted party or outsources the
+training procedure to such a party that we refer to as the ad-
+versary. The adversary provides the user with a model that
+performs well on an unseen validation set, but produces a
+pre-defined class label in the presence of an attacker-defined
+trigger called the backdoor trigger. The association between
+the backdoor trigger and the attacker-specified label is cre-
+ated by training the DNN on poisoned training samples,
+which are samples polluted by the attacker’s trigger [39].
+In poisoned-label attacks, unlike clean-label attacks, the at-
+tacker also switches the label of the poisoned samples to the
+intended target label.
+Considerable attention has been paid to explore back-
+door attacks and defenses for 2D image classification mod-
+els [5,22,25]. However, little attention has been paid to ex-
+ploring backdoor attacks and defenses against video action
+recognition models. The disappointing conclusion uncov-
+ered by [87] regarding the limited effectiveness of image
+backdoor attacks on videos stunted further development of
+video backdoor attacks. Unfortunately, the attacks consid-
+ered in [87] were limited to only visible patch-based clean-
+label attacks. Moreover, [87] directly adopted the 2D back-
+door attack threat model without incorporating important
+video-specific considerations.
+To this end, and as opposed to [87]. we first revisit and
+revise the commonly adopted 2D poisoned-label backdoor
+threat model by incorporating additional constraints that are
+inherently imposed by video systems.
+These constraints
+arise due to the presence of the temporal dimension. We
+then explore two ways to extend image backdoor attacks to
+incorporate the temporal dimension into the attack to enable
+more video-specific backdoor attacks. In particular, image
+backdoor attacks could be either extended statically by ap-
+plying the same attack to each frame of the video or dynam-
+ically by adjusting the attack parameters differently for each
+frame. Then, three novel natural video backdoor attacks are
+presented to highlight the seriousness of the risks associ-
+ated with backdoor attacks in the video domain. We then
+test the attacked models against three 2D backdoor defenses
+and discuss the reason behind the failure of those methods.
+arXiv:2301.00986v1  [cs.CV]  3 Jan 2023
+
+We also study, for the first time, audiovisual backdoor at-
+tacks, where we ablate the importance and contribution of
+each modality on the performance of the attack for both late
+and early fusion settings. We show that attacking a single
+modality is enough to achieve a high attack success rate.
+Contributions. Our contributions are twofold. (1) We re-
+visit the traditional backdoor attack threat model and incor-
+porate video-related aspects, such as video subsampling and
+spatial cropping, into the model. We also extend existing
+image backdoor attacks to the video domain in two differ-
+ent ways, statically and dynamically, after which we pro-
+pose three novel natural video backdoor attacks. Through
+extensive experiments, we provide evidence that the previ-
+ous perception of image backdoor attacks in the video do-
+main is not necessarily true, especially in the poisoned-label
+attack setup. (2) To the best of our knowledge, this work is
+the first to investigate audiovisual backdoor attacks against
+video action recognition models.
+2. Related Work
+Backdoor Attacks. Backdoor attacks were first introduced
+in [22]. The attack, called BadNet, was based on adding a
+patch to the corner of a subset of training images to create a
+backdoor that could be triggered by the attacker at will. Fol-
+lowing BadNet, [44] proposed optimizing for the values of
+the patch to obtain a more effective backdoor attack. Shortly
+after the development of patch-based backdoor attacks, the
+community realized the importance of adding an invisibility
+constraint to the design of backdoor triggers to bypass any
+human inspection. Works such as [9] proposed blending
+the backdoor trigger with the image rather than stamping
+it. [37] generated backdoor attacks using the least signifi-
+cant bit algorithm. [52] generated warping fields to warp the
+image content as a backdoor trigger. [14] went one step fur-
+ther and designed learnable transformations to generate op-
+timal backdoor triggers. After many attacks were proposed
+in the spatial domain [10, 37, 40, 45, 56, 57, 67, 72, 75], and
+others in the latent representation domain [13,53,80,88,91],
+[19, 25, 71, 82, 84] proposed to switch attention to the fre-
+quency domain. [25] utilized frequency heatmaps proposed
+in [81] to create backdoor attacks that target the most sen-
+sitive frequency components of the network. [19] proposed
+blending low frequency content from a trigger image with
+training images as a poisoning technique. In our work, we
+extend the 2D backdoor threat model to the video domain
+by incorporating video-related aspects into it. We also ex-
+tend five image backdoor attacks into the video domain and
+propose three natural video backdoor attacks.
+Backdoor Defenses. Backdoor attack literature was im-
+mediately opposed by various defenses.
+Backdoor de-
+fenses are generally of five types:
+preprocessing-based
+[12,47,55], model reconstruction-based [38,42,74,83,89],
+trigger synthesis-based [23,24,29,43,54,58,62,68], model
+diagonsis-based [15, 34, 46, 76, 90], and sample-filtering
+based [8, 20, 26, 30, 61, 63]. Early backdoor defenses such
+as [68] hypothesized that backdoor attacks create a short-
+cut between all samples and the poisoned class. Based on
+that, they solved an optimization problem to find whether a
+trigger of an abnormally small norm exists that would flip
+all samples to one label. Later, multiple improved itera-
+tions of this method were proposed, such as [23, 43, 83].
+Fine pruning [42] suggested that the backdoor is triggered
+by particular neurons that are dormant in the absence of
+the trigger. Therefore, the authors proposed pruning the
+least active neurons on clean samples. STRIP [20] showed
+that blending clean samples with other clean samples would
+yield a higher entropy compared to when clean images are
+blended with poisoned samples. Activation clustering [8]
+uses KMeans to cluster the activations of an inspection, a
+potentially poisoned data set, into two clusters. A large sil-
+houette distance between the two clusters would uncover
+the poisoned samples. In our work, we show that current
+image backdoor attacks have limited effectiveness in de-
+fending against backdoor attacks in the video domain, es-
+pecially against the proposed natural video attacks.
+Video Action Recognition. Video action recognition mod-
+els, which only leverage the raw frames of a video, can be
+categorized into two categories, CNN-based networks and
+transformer-based networks. 2D CNN-based methods are
+built on top of pretrained image recognition networks with
+well-designed modules to capture the temporal relationship
+between multiple frames [41,49,69,70]. Those methods are
+computationally efficient as they use 2D convolutional ker-
+nels. To learn stronger spatial-temporal representations, 3D
+CNN-based methods were proposed. These methods utilize
+3D kernels to jointly leverage the spatio-temporal context
+within a video clip [17, 18, 64, 65]. To better initialize the
+network, I3D [7] inflated the weights of 2D pretrained im-
+age recognition models to adapt them to 3D CNNs. Real-
+izing the importance of computational efficiency, S3D [78]
+and R(2+1)D [66] proposed to disentangle spatial and tem-
+poral convolutions to reduce computational cost. Recently,
+transformer-based action recognition models were able to
+achieve better performance in large training data sets com-
+pared to CNN-based models, e.g. [4,6,16,48]. In this work,
+we test backdoor attacks against three action recognition
+architectures, namely I3D, SlowFast, and TSM.
+Audiovisual Action Recognition. In addition to frames, a
+line of action recognition models [1,27,28,51] has used the
+accompanying audio to better understand activities such as
+“playing music” or “washing dishes”. To take advantage of
+existing CNN and transformer-based models, the Log-Mel
+spectrogram was introduced to convert audio data from a
+non-structured signal into a 2D representation in time and
+frequency usable by these models [2,3,35,77]. Current au-
+diovisual action recognition methods are divided into two
+
+Figure 1. Traditional Backdoor Attack Pipeline. After selecting a backdoor trigger and a target label, the attacker poisons a subset of
+the training data referred to as the poisoned dataset (Dp). The label of the poisoned dataset is fixed to a target poisoning label specified by
+the attacker. The attacker trains jointly on clean (non-poisoned) samples (Dc) and poisoned samples leading to a backdoored model, which
+outputs the target label in the presence of the backdoor trigger.
+categories based on when the audio and visual signals are
+merged in the recognition pipeline: early fusion and late fu-
+sion. Early fusion combines features before classification,
+which can better capture features [32,77]. The disadvantage
+of early fusion is that there is a higher risk of overfitting to
+the training data [59]. Late fusion, on the other hand, treats
+the video and audio networks separately, and the predictions
+of each network are carried out independently, after which
+the logits are aggregated to make a final prediction [21]. For
+the first time, we test backdoor attacks against audiovisual
+action recognition networks in both late and early fusion
+setups.
+3. Video Backdoor Attacks
+3.1. The Traditional Threat Model
+The commonly adopted threat model for backdoor at-
+tacks dates back to the works that studied those attacks
+against 2D image classification models [22]. The victim
+outsources the training process to a trainer who is given ac-
+cess to both the victim’s training data and the network ar-
+chitecture. The victim only accepts the model provided by
+the trainer if it performs well on the victim’s private val-
+idation set. The attacker aims to maximize the effective-
+ness of the embedded backdoor attack [39]. We refer to
+the model’s performance on the validation set as clean data
+accuracy (CDA). The effectiveness of the backdoor attack
+is measured by the attack success rate (ASR), which is de-
+fined as the percentage of test examples not labeled as the
+target class that are classified as the target class when the
+backdoor pattern is applied. To achieve this goal, the at-
+tacker applies a backdoor trigger to a subset of the train-
+ing images and then, in the poisoned-label setup, switches
+the labels of those images to a target class of choice before
+training begins. A more powerful backdoor attack is one
+that is visually imperceptible (usually measured in terms of
+ℓ2/ℓ∞-norm, PSNR, SSIM, or LPIPS) but achieves both a
+high CDA and a high ASR. This is summarized in Figure 1.
+More formally, we denote the classifier which is param-
+eterized by θ as fθ : X → Y. It maps the input x ∈ X,
+such as images or videos, to class labels y ∈ Y.
+Let
+Gη : X → X indicate an attacker-specific poisoned im-
+age generator that is parameterized by some trigger-specific
+parameters η. The generator may be image-dependent. Fi-
+nally, let S : Y → Y be an attacker-specified label shifting
+function. In our case, we consider the scenario in which the
+attacker is trying to flip all the labels into one particular la-
+bel, i.e. S : Y → t, where t ∈ Y is an attacker-specified
+label that will be activated in the presence of the backdoor
+trigger. Let D = {(xi, yi)}N
+i=1 indicate the training dataset.
+The attacker splits D into two subsets, a clean subset Dc and
+a poisoned subset Dp, whose images are poisoned by Gη
+and labels are poisoned by S. The poisoning rate is the ra-
+tio α = |Dp|
+|D| , generally a lower poisoning rate is associated
+with a higher clean data accuracy. The attacker typically
+trains the network by minimizing the cross-entropy loss on
+Dc∪Dp, i.e. minimizes E(x,y)∼Dc∪Dp[LCE(fθ(x), y)]. The
+attacker aims to achieve high accuracy on the user’s valida-
+tion set Dval while being able to trigger the poisoned-label,
+t, in the presence of the backdoor trigger, i.e. fθ(Gη(x)) =
+t, ∀x ∈ X (ideally).
+3.2. From Images to Videos
+Unlike images, videos have an additional dimension, the
+temporal dimension. This dimension introduces new rules
+to the game between the attacker and the victim.
+More
+
+Settings
+Training Stage
+Inference Stage
+"Eat"
+"Jump'
+fe
+fe
+Poisoned
+Network
+Target Label = "Eat"
+Backdoor Trigger
+Clean Samples (Dc)
+Poisoned Video (Gn(α))
+(α)
+Label = "Eat"
+Clean Video
+Ground Truth LabelFigure 2. Static vs Dynamic Backdoor Attacks. Static backdoor attacks apply the same trigger across all frames along the temporal
+dimension. On the other hand, dynamic attacks apply a different trigger per frame along the temporal dimension.
+precisely, the attacker now has an additional dimension to
+hide the backdoor trigger, leading to a higher level of im-
+perceptibility.
+The backdoor attack could be applied to
+all the frames or a subset of the frames statically, i.e. the
+same trigger is applied to each frame, or dynamically, i.e.
+a different trigger is applied to each frame. On the other
+hand, the testing pipeline now imposes harsher conditions
+against the backdoor attack. Video recognition models tend
+to test the model on multiple sub-sampled clips with vari-
+ous crops [7,18,41] which might, in turn, destroy the back-
+door trigger.
+For example, if the trigger is applied to a
+single frame, it might not be sampled, and if the trigger
+is applied to the corner of the image, it might be cropped
+out. The threat model presented in Subsection 3.1 was di-
+rectly adopted in [87], which to the best of our knowledge,
+is the only previous work that considered backdoor attacks
+for video action recognition.
+Our work sheds light on the aforementioned video-
+related aspects. In Section 4.2, we show the effect of the
+number of frames poisoned on CDA and ASR. We also
+show how existing 2D methods could be extended both stat-
+ically and dynamically to suit the video domain. For exam-
+ple, BadNet [22] applies a fixed patch as a backdoor trigger.
+The patch could be applied statically using the same pixel
+values and the same position along the temporal dimension
+or applied dynamically by changing the position and possi-
+bly the pixel values of the patch for each frame. Figure 2
+shows a BadNet attack when applied in a static and dynamic
+way. Additionally, we show how simple yet natural video
+“artifacts” could be used as backdoor triggers. More specif-
+ically, lag in a video, motion blur, and compression glitches
+could all be used as naturally occurring backdoor triggers.
+3.3. Audiovisual Backdoor Attacks
+Videos are naturally accompanied by audio signals. Sim-
+ilarly to how the video modality could be attacked, the audio
+signal could also be attacked. The interesting question that
+arises is how backdoor attacks would perform in a multi-
+modal setup. In the experiments of Section 4.4, we answer
+the following questions: (1) What is the effect of having two
+attacked modalities on CDA and ASR?; (2) What happens
+if only one modality is attacked and the other is left clean?;
+(3) What is the difference in performance between late and
+early fusion in terms of CDA and ASR?
+4. Experiments
+4.1. Experimental Settings
+Datasets. We consider three standard benchmark datasets
+used in video action recognition: UCF-101 [60], HMDB-
+51 [36], and Kinetics-Sounds [31]. Kinetics-Sounds is a
+subset of Kinetics400 that contains classes that can be clas-
+sified from the audio signal, i.e. classes where audio is use-
+ful for action recognition [2]. Kinetics-Sounds is particu-
+larly interesting for Sections 4.3 and 4.4, where we explore
+backdoor attacks against audio and audiovisual classifiers.
+Network Architectures. Following common practice, for
+the visual modality, we use a dense sampling strategy to
+sub-sample 32 frames per video to fine-tune a pretrained
+I3D network on the target dataset [7]. In Section 4.2, we
+also show results using TSM [41] and SlowFast [18] net-
+works. All three models adopt ResNet-50 as the backbone
+and are pretrained on Kinetics-400. Similarly to [2], for
+the audio modality, a ResNet-18 is trained from scratch on
+Mel-Spectrograms composed of 80 Mel bands sub-sampled
+temporally to a fixed length of 256.
+Attack Setting.
+For the video modality, we study and
+extend the following image-based backdoor attacks to the
+video domain: BadNet [22], Blend [9], SIG [5], WaNet
+[52], and FTrojan [71]. We also explore three additional
+natural video backdoor attacks.
+For the audio modality,
+we consider two attacks: sine attack and high-frequency
+noise attack, both of which we explain later.
+Following
+[22,25,52], the target class is arbitrarily set to the first class
+
+t=0
+t=1
+t=12
+t=13
+t=14
+t=N-1
+t=N
+Static
+DynamicFigure 3. Visualization of 2D Backdoor Attacks. Image backdoor attacks mainly differ according to the backdoor trigger used to poison
+the training samples. They could be extended either statically or dynamically based on how the attack is applied across the frames.
+of each data set (class 0), and the poisoning rate is set to
+10%. Unless otherwise stated, the considered image back-
+door attacks poison all frames of the sampled clips during
+training and evaluation.
+Evaluation Metrics. As is commonly done in the back-
+door literature, we evaluate the performance of the model
+using clean data accuracy (CDA) and attack success rate
+(ASR) explained in Section 3. CDA represents the usual
+validation/test accuracy on an unseen dataset hence mea-
+suring the generalizability of the model. On the other hand,
+ASR measures the effectiveness of the attack when the poi-
+son is applied to the validation/test set.
+In addition, we
+test the attacked models against some of the early 2D back-
+door defenses, more precisely against activation clustering
+(AC) [8], STRIP [20], and pruning [42].
+Implementation Details. Our method is built on MMAc-
+tion2 library [11], and follows their default training config-
+urations and testing protocols, except for the learning rate
+and the number of training epochs (check Supplementary).
+All experiments were run using 4 NVIDIA A100 GPUs.
+4.2. Video Backdoor Attacks
+Extending Image Backdoor Attacks to the Video Do-
+main. As mentioned in Section 3.2, image backdoor attacks
+could be extended either statically by applying an attack in
+the same way across all frames or dynamically by adjusting
+the attack parameters for different frames. We consider five
+attacks that differ according to the applied backdoor trig-
+ger. BadNet applies a patch as a trigger, Blend blends a
+trigger image to the original image, SIG superimposes a si-
+nusoidal trigger to the image, WaNet warps the content of
+the image, and FTrojan poisons a high- and mid- frequency
+component in the discrete cosine transform (DCT). Figure 3
+visualizes all five attacks on the same video frame. Each of
+the considered methods could be extended dynamically as
+follows: BadNet: change the patch location for each frame;
+Blend: blend a uniform noise that is different per frame;
+SIG: change the frequency of the sine component superim-
+posed with each frame; WaNet: generate a different warp-
+ing field for each frame; FTrojan: select a different DCT
+basis to perturb at each frame. Note that Blend and FTro-
+jan are generally imperceptible. Visualizations and saliency
+UCF101
+HMDB51
+KineticsSound
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+93.95
+-
+69.59
+-
+81.41
+-
+BadNet
+93.95
+99.63
+69.35
+98.89
+82.97
+99.09
+Blend
+94.29
+99.26
+68.37
+86.73
+82.12
+97.54
+SIG
+93.97
+99.97
+68.50
+99.80
+82.84
+99.87
+WaNet
+94.05
+99.84
+68.95
+99.61
+82.38
+99.09
+FTrojan
+94.16
+99.34
+68.10
+97.52
+82.45
+97.86
+Table 1. Statically Extended 2D Backdoor Attacks. Statically
+extending 2D backdoor attacks to the video domain leads to high
+CDA and ASR across all three considered datasets.
+UCF101
+HMDB51
+KineticsSound
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+93.95
+-
+69.59
+-
+81.41
+-
+BadNet
+94.11
+99.97
+69.08
+99.54
+82.25
+99.74
+Blend
+94.21
+99.44
+67.03
+95.95
+81.67
+95.79
+SIG
+94.24
+100.00
+68.63
+100.00
+82.84
+100.00
+WaNet
+94.29
+99.79
+69.22
+99.80
+82.25
+99.61
+FTrojan
+94.16
+99.34
+67.19
+98.69
+82.25
+95.27
+Table 2. Dynamically Extended 2D Backdoor Attacks. Dynam-
+ically extending 2D backdoor attacks to the video domain leads to
+high CDA and ASR across all three considered datasets.
+maps for each attack are found in the Supplementary.
+Tables 1 and 2 show the CDA and ASR of the I3D mod-
+els attacked using various backdoor attacks on UCF-101,
+HMDB-51, and Kinetics-Sounds. Contrary to the conclu-
+sion presented in [87], we find that backdoor attacks are
+actually highly effective in the video domain. The CDA
+of the attacked models is very similar to that of the clean
+unattacked model (baseline), surpassing it in some cases.
+Extending attacks dynamically, almost always, improves
+CDA and ASR compared to extending them statically.
+Natural Video Backdoors. A more interesting attack is
+one that seems natural and could bypass human inspec-
+tion [50, 73, 79, 86]. There are several natural “glitches”
+that occur in the video domain and that one could exploit
+to design a natural backdoor attack. For example, videos
+might contain some frame lag, motion blur, video compres-
+sion corruptions, camera focus/defocus, etc. In Table 3, we
+report the CDA and ASR of three natural backdoor attacks:
+
+Clean
+BadNet
+Blend
+SIG
+WaNet
+FTrojanUCF101
+HMDB51
+KineticsSound
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+93.95
+-
+69.59
+-
+81.41
+-
+Frame Lag
+92.94
+97.20
+68.04
+98.76
+82.51
+98.19
+Video Corrupt.
+94.26
+99.87
+69.22
+99.22
+81.74
+98.51
+Motion Blur
+93.97
+99.92
+68.17
+97.52
+82.19
+99.22
+Table 3.
+Natural Video Backdoor Attacks.
+Natural attacks
+against video action recognition models could achieve high CDA
+and ASR while looking completely natural to human inspection.
+SlowFast
+TSM
+CDA(%)
+ASR(%)
+CDA(%)
+ASR(%)
+Baseline
+96.72
+-
+94.77
+-
+BadNet
+96.64
+99.47
+94.69
+97.78
+SIG
+96.70
+99.97
+94.77
+99.47
+FTrojan
+96.25
+98.52
+94.21
+100.00
+Frame Lag
+96.43
+99.97
+94.63
+97.96
+Video Corruption
+96.54
+99.76
+95.08
+98.97
+Motion Blur
+96.46
+99.55
+94.50
+99.39
+Table 4.
+Video Backdoor Attacks Against Different Archi-
+tectures (UCF-101). When tested against network architectures
+other than I3D such as TSM and SlowFast, both image and natural
+backdoor attacks can still achieve high CDA and high ASR.
+frame lag (lagging video), video compression glitch (which
+we refer to as Video Corruption), and motion blur. Interest-
+ingly, these attacks could achieve both high clean data ac-
+curacy and high attack success rate. It is worth noting that
+for frame lag, a two-frame lag is used for UCF-101 and a
+three-frame lag is used for HMDB-51 and Kinetics-Sounds.
+More details are provided in the Supplementary.
+Attacks Against Different Architectures. So far, all at-
+tacks have been experimented with against an I3D network.
+To further explore the behavior of backdoor attacks against
+other video recognition models, we test a subset of the con-
+sidered attacks against a 2D based model, TSM, and another
+3D based model, SlowFast, on UCF-101. Table 4 shows
+that all the aforementioned backdoor attacks perform sig-
+nificantly well in terms of CDA and ASR against both TSM
+and SlowFast architectures. Note that even though TSM is
+a 2D based model, our proposed natural video backdoor at-
+tacks still succeed in attacking it.
+Recommendations for Video Backdoor Attacks. As men-
+tioned in Section 3.2, the attacker must select a number of
+frames to poison per video, keeping in mind that the video
+will be sub-sampled and randomly cropped during evalua-
+tion. Since the attacker is the one who trained the network in
+the first place, he/she has access to the processing pipeline
+and could exploit this during the attack. For example, if
+video processing involves sub-sampling the video into clips
+of 32 frames and cropping the frames into 224×224 crops,
+the attacker could pass to the network an attacked video of
+a temporal length of 32 frames and a spatial size 224×224,
+Figure 4. Effect of the Number of Poisoned Frames (UCF-101).
+Different colors refer to different number of frames poisoned dur-
+ing the training of the attacked model. Training the model with
+a single poisoned frame performs best for various choices of the
+number of frames poisoned during evaluation.
+Frame
+Lag
+Motion
+Blur
+SIG
+BadNet
+FTrojan
+Elimination Rate(%)
+0.00
+0.00
+34.21
+33.77
+34.12
+Sacrifice Rate(%)
+13.08
+12.82
+15.17
+14.25
+13.00
+Table 5. Activation Clustering Defense (UCF-101). Whereas
+Activation Clustering provides partial success in defending against
+image backdoor attacks, it fails completely against natural attacks.
+hence bypassing sub-sampling and cropping. However, a
+system could force the user to input a video of a partic-
+ular length, possibly greater than the length of the sub-
+sampled clips. This raises an important question regarding
+how many frames the attacker should poison. Clearly, the
+smaller the number of frames the attacker poisons, the less
+detectable the attack is, but does the attack remain effective?
+In Figure 4, we show the attack success rate of backdoor-
+attacked models trained on clips of 1, 8, 16, and 32 frames,
+and a randomly sampled number of poisoned frames (out of
+32 total frames) when evaluated on clips of 1, 8, 16, and 32
+poisoned frames (out of 32 total frames). Random refers to
+training on a varying number of poisoned frames per clip.
+Note that training the model against the worst-case scenario
+(single frame), which mimics the case where only one of
+the poisoned frames is sub-sampled, provides the best guar-
+antees for achieving a high attack success rate.
+Defenses Against Video Backdoor Attacks. We explore
+the effect of extending some of the existing 2D backdoor
+defenses against video backdoor attacks.
+Optimization-
+based defenses are extremely costly when extended to the
+video domain. For example, Neural Cleanse (NC) [68], I-
+BAU [83], and TABOR [23] involve a trigger reconstruc-
+tion phase. The trigger space is now bigger in the presence
+of the temporal dimension, and therefore, instead of opti-
+mizing for a 224×224×3 trigger, the defender has to search
+for a 32×224×224×3 trigger (assuming 32 frame clips are
+used), which is both costly and hard to solve. The attacker
+has the spatial and temporal dimensions to design and em-
+
+BadNet
+SIG
+100
+75
+# Poisoned Frames
+ASR(%)
+(Training)
+1
+50
+8
+16
+25
+32
+Random
+0
+8
+16
+32 1
+1
+8
+16
+32
+# Poisoned Frames
+(EvaluationFigure 5. STRIP Defense (UCF-101). Whereas the entropy of
+image backdoor attacks is very low compared to that of clean sam-
+ples, the proposed natural backdoor attacks have a natural distri-
+bution of entropies similar to that of clean samples.
+Figure 6. Pruning Defense (Kinetics-Sounds). Pruning is com-
+pletely ineffective against image backdoor attacks extended to the
+video domain and natural video backdoor attacks. Even though
+the clean accuracy has dropped to random, the attack success rate
+is maintained at very high levels.
+bed their attack in, and, therefore, reverse engineering the
+trigger is quite hard.
+We consider three well-known defenses that introduce no
+computational overhead when adopted to the video domain,
+namely Activation Cluster (AC) [8], STRIP [20], and prun-
+ing [42]. AC computes the activations of a neural network
+on clean samples (from the test set) and an inspection set
+of interest which may be poisoned. AC then applies PCA
+to reduce the dimension of the activations, after which the
+projected activations are clustered into two classes and com-
+pared to the activations of the clean set. STRIP blends clean
+samples with the samples of a possibly poisoned inspec-
+tion set. The entropy of the predicted probabilities is then
+checked for any abnormalities. Unlike clean samples, poi-
+soned samples tend to have a low entropy. Pruning suggests
+that the backdoor is usually embedded in particular neurons
+Baseline
+Sine Attack
+High Frequency Attack
+CDA(%)
+49.21
+47.21
+47.61
+ASR(%)
+-
+96.36
+95.96
+Table 6. Audio Backdoor Attacks (Kinetics-Sounds). Both sine
+attack and the high-frequency band attack perform similarly to
+baseline in terms of CDA while being able to achieve high ASR.
+in the network that are only activated in the presence of the
+trigger. Therefore, those neurons are supposed to be dor-
+mant as far as the test set samples, i.e. clean samples, are
+concerned. This allows us to detect and prune those dor-
+mant neurons to eliminate the backdoor. Table 5 shows the
+elimination and sacrifice rates of AC when applied against
+some of the considered attacks. The elimination rate refers
+to the ratio of poisoned samples correctly detected as poi-
+soned to the total number of poisoned samples, whereas the
+sacrifice rate refers to the ratio of clean samples incorrectly
+detected as poisoned to the total number of clean samples.
+Whereas AC has partial success in defending against image
+backdoor attacks, it fails completely against the proposed
+natural backdoor attacks. Figure 5 shows that the entropy
+of the clean and poisoned samples of the proposed natural
+attacks is very similar and therefore could evade the STRIP
+defense, while BadNet and FTrojan are detectable. Finally,
+Figure 6 shows that pruning the least active neurons causes
+a reduction in CDA without reducing ASR. This is observed
+not only for the natural attacks, but also for the extended im-
+age backdoor attacks, hinting that image backdoor defenses
+are not effective in the video domain.
+4.3. Audio Backdoor Attacks
+Attacks proposed against audio networks have been lim-
+ited to adding a low-volume one-hot-spectrum noise in the
+frequency domain, which leaves highly visible artifacts in
+the spectrogram [85] or adding a human non-audible com-
+ponent [33], f < 20Hz or f > 20kHz, which is non-
+realistic, since spectrograms usually filter out those frequen-
+cies. We consider two attacks against the Kinetics-Sounds
+dataset; the first is to add a low-amplitude sine wave com-
+ponent with f = 800Hz to the audio signal, and the second
+is to add band-limited noise 5kHz < f < 6kHz. The spec-
+trograms and the absolute difference between the attacked
+spectrograms and the clean spectrogram are shown in Fig-
+ure 7. Since no clear artifacts are observed in the spectro-
+grams, human inspection fails to label the spectrograms as
+attacked. The CDA and ASR rates of the backdoor-attacked
+models for both attacks are shown in Table 6. The attacks
+achieve a relatively high ASR.
+4.4. Audiovisual Backdoor Attacks
+Now, we combine video and audio attacks to build a
+multi-modal audiovisual backdoor attack. The way we do
+
+BadNet
+FTrojan
+1500
+1500
+Poisoned
+Clean
+1000
+1000
+500
+500
+0
+0
+2
+4
+0
+2
+4
+Frame Lag
+Motion Blur
+600
+600
+400
+400
+200
+200
+0
+0
+0
+2
+4
+0
+2
+4
+EntropyBadNet
+Frame Lag
+100
+ASR
+ASR
+CDA
+CDA
+Accuracy(%)
+75
+50
+25
+0
+0
+25
+50
+75
+100
+0
+25
+50
+75
+100
+Percentage Pruned (%Late Fusion
+Early Fusion
+Clean Audio
+Sine Attack
+High Freq. Attack
+Clean Audio
+Sine Attack
+High Freq. Attack
+Clean Video
+80.25 / -
+81.74 / 70.98
+80.96 / 77.91
+84.72 / -
+83.48 / 92.23
+83.94 / 93.72
+BadNet
+77.33 / 66.97
+78.63 / 99.74
+77.33 / 99.87
+87.50 / 99.29
+85.10 / 99.87
+85.75 / 100.00
+Blend
+79.60 / 75.06
+80.76 / 99.68
+79.08 / 99.61
+86.08 / 98.19
+83.55 / 99.81
+85.43 / 99.87
+SIG
+78.50 / 68.33
+80.12 / 99.87
+79.02 / 100.00
+86.92 / 99.81
+84.97 / 100.00
+85.95 / 100.00
+WaNet
+77.66 / 68.39
+79.79 / 99.94
+79.02 / 99.94
+86.46 / 98.96
+84.97 / 100.00
+85.88 / 100.00
+FTrojan
+79.66 / 67.16
+80.76 / 99.48
+79.99 / 99.29
+86.08 / 98.58
+84.65 / 99.94
+85.49 / 100.00
+Frame Lag
+79.08 / 63.41
+80.57 / 99.74
+79.47 / 99.87
+86.08 / 98.19
+84.59 / 99.94
+84.65 / 100.00
+Video Corruption
+78.11 / 64.57
+78.24 / 99.68
+77.66 / 99.94
+86.59 / 99.29
+84.59 / 100.00
+85.43 / 100.00
+Motion Blur
+79.79 / 69.24
+80.70 / 99.68
+79.86 / 99.94
+86.40 / 98.58
+84.65 / 100.00
+85.62 / 100.00
+Table 7. Audiovisual Backdoor Attacks (Kinetics-Sounds). The entries in the table report the CDA(%)/ASR(%) of attacking late and
+early fused audiovisual networks. When a single modality is attacked, late fusion has a low ASR compared to early fusion. When both
+modalities are attacked, the ASR of both late and early fusion are high.
+Figure 7. Clean and Attacked Audio Spectrograms. The uti-
+lized audio backdoor attacks are not only audibly imperceptible
+but also leave no perceptible artifacts in the Mel spectrogram. The
+spectrogram of each attack is followed by the absolute difference
+of the attacked spectrogram with the clean one.
+it is by taking our attacked models from Sections 4.2 and
+4.3 and applying early or late fusion. For early fusion, we
+extract video and audio features using our trained audio and
+video backbones, and we then train a classifier on the con-
+catenation of the features. In late fusion, the video and au-
+dio networks predict independently on the input, and then
+the individual logits are aggregated to produce the final pre-
+diction. To answer the three questions posed in Section 3.3,
+we run experiments in which both modalities are attacked
+and others in which only a single modality is attacked for
+both early and late fusion setups (Table 7). We summarize
+the results as follows. (1) Attacking two modalities con-
+sistently improves ASR and even CDA in some cases. (2)
+Attacking a single modality is good enough to achieve a
+high ASR in the case of early fusion but not late fusion.
+(3) Early fusion enables the best of both worlds for the at-
+tacker, namely, a high CDA and an almost perfect ASR. On
+the other hand, late fusion experiences some serious drops
+in ASR in the unimodal attack setup. An interesting find-
+ing in these experiments is the following: if the outsourcer
+has the option to outsource the most expensive modality,
+training wise, while training other modalities in-house, ap-
+plying late fusion could be used as a defense mechanism,
+especially in the presence of more clean modalities.
+5. Conclusion
+Backdoor attacks present a serious and exploitable vul-
+nerability against both unimodal and multi-modal video
+action recognition models. We showed how existing im-
+age backdoor attacks could be extended either statically
+or dynamically to develop powerful backdoor attacks that
+achieve both a high clean data accuracy and a high attack
+success rate. Besides existing image backdoor attacks, there
+exists a set of natural video backdoor attacks, such as mo-
+tion blur and frame lag, that are resilient to existing image
+backdoor defenses. Given that videos are usually accom-
+panied by audio, we showed two ways in which one could
+attack audio classifiers in a human inaudible manner. The
+
+Clean Spectrogram
+80
+40
+0
+Sine Attack
+80
+40
+0
+Mel Frequency
+80
+40
+0
+High Frequency Attack
+80
+40
+0
+80
+40
+0
+0
+50
+100
+150
+200
+250
+300
+Frameattacked video and audio models are then used to train an
+audiovisual action recognition model by applying both early
+and late fusion. Different combinations of poisoned modal-
+ities are tested, concluding that: (1) poisoning two modal-
+ities could achieve extremely high attack success rates in
+both late and early fusion settings, and (2) if a single modal-
+ity is poisoned, unlike early fusion, late fusion could reduce
+the effectiveness of the backdoor. We hope that our work
+reignites the attention of the community towards exploring
+backdoor attacks and defenses in the video domain.
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+

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+A Safety Framework for Flow Decomposition Problems via
+Integer Linear Programming
+Fernando H. C. Dias1,⋆[0000−0002−6398−919X], Manuel C´aceres1,⋆[0000−0003−0235−6951], Lucia
+Williams2,⋆[0000−0003−3785−0247], Brendan Mumey2,⋆⋆[0000−0001−7151−2124], and
+Alexandru I. Tomescu1,⋆⋆[0000−0002−5747−8350]
+1 Department of Computer Science, University of Helsinki, Finland
+{fernando.cunhadias,manuel.caceres,alexandru.tomescu}@helsinki.fi
+2 School of Computing, Montana State University, Bozeman, MT, USA
+{lucia.williams,brendan.mumey}@montana.edu
+Abstract. Many important problems in Bioinformatics (e.g., assembly or multi-assembly) admit mul-
+tiple solutions, while the final objective is to report only one. A common approach to deal with this
+uncertainty is finding safe partial solutions (e.g., contigs) which are common to all solutions. Previous
+research on safety has focused on polynomially-time solvable problems, whereas many successful and
+natural models are NP-hard to solve, leaving a lack of “safety tools” for such problems. We propose
+the first method for computing all safe solutions for an NP-hard problem, minimum flow decomposi-
+tion. We obtain our results by developing a “safety test” for paths based on a general Integer Linear
+Programming (ILP) formulation. Moreover, we provide implementations with practical optimizations
+aimed to reduce the total ILP time, the most efficient of these being based on a recursive group-testing
+procedure.
+Results: Experimental results on the transcriptome datasets of Shao and Kingsford (TCBB, 2017)
+show that all safe paths for minimum flow decompositions correctly recover up to 90% of the full RNA
+transcripts, which is at least 25% more than previously known safe paths, such as (C´aceres et al. TCBB,
+2021), (Zheng et al., RECOMB 2021), (Khan et al., RECOMB 2022, ESA 2022). Moreover, despite the
+NP-hardness of the problem, we can report all safe paths for 99.8% of the over 27,000 non-trivial graphs
+of this dataset in only 1.5 hours. Our results suggest that, on perfect data, there is less ambiguity than
+thought in the notoriously hard RNA assembly problem.
+Availability: https://github.com/algbio/mfd-safety
+Contact: alexandru.tomescu@helsinki.fi
+Keywords: RNA assembly · Network flow · Flow decomposition · Integer linear programming · Safety
+⋆ Shared first-author contribution
+⋆⋆ Shared last-author contribution
+arXiv:2301.13245v1  [cs.DS]  30 Jan 2023
+
+1
+Introduction
+In real-world scenarios where an unknown object needs to be discovered from the input data, we would like
+to formulate a computational problem loosely enough so that the unknown object is indeed a solution to
+the problem, but also tightly enough so that the problem does not admit many other solutions. However,
+this goal is difficult in practice, and indeed, various commonly used problem formulations in Bioinformatics
+still admit many solutions. While a naive approach is to just exhaustively enumerate all these solutions, a
+more practical approach is to report only those sub-solutions (or partial solutions) that are common to all
+solutions to the problem.
+In the graph theory community such sub-solutions have been called persistent [14,21], and in the Bioin-
+formatics community reliable [54], or more recently, safe [51]. The study of safe sub-solutions started in
+Bioinformatics in the 1990’s [54,11,37] with those amino-acid pairs that are common to all optimal and
+suboptimal alignments of two protein sequences.
+In the genome assembly community, the notion of contig, namely a string that is guaranteed to appear in
+any possible assembly of the reads, is at the core of most genome assemblers. This approach originated in 1995
+with the notion of unitigs [25] (non-branching paths in an assembly graph), which were progressively [42,6]
+generalized to paths made up of a prefix of nodes with in-degree one followed by nodes with out-degree
+one [35,24,29] (also called extended unitigs, or Y-to-V contigs).
+Later, [51] formalized all such types of contigs as those safe strings that appear in all solutions to a
+genome assembly problem formulation, expressed as a certain type of walk in a graph. [10,9] proposed more
+efficient and unifying safety algorithms for several types of graph walks. [45] recently studied the safety of
+contigs produced by state-of-the-art genome assemblers on real data.
+Analogous studies were recently made also for multi-assembly problems, where several related genomic
+sequences need to be assembled from a sample of mixed reads. [8] studied safe paths that appear in all
+constrained path covers of a directed acyclic graph (DAG). Zheng, Ma and Kingsford studied the more
+practical setting of a network flow in a DAG by finding those paths that appear in any flow decomposition
+of the given network flow, under a probabilistic framework [34], or a combinatorial framework [58].3 [27]
+presented a simple characterization of safe paths appearing in any flow decomposition of a given acyclic
+network flow, leading to a more efficient algorithm than the one of [58], and further optimized by [28].
+Motivation. Despite the significant progress in obtaining safe algorithms for a range of different appli-
+cations, current safe algorithms are limited to problems where computing a solution itself is achievable in
+polynomial time. However, many natural problems are NP-hard, and safe algorithms for such problems are
+fully missing. Apart from the theoretical interest, usually such NP-hard problems correspond to restrictions
+of easier (polynomially-computable) problems, and thus by definition, also have longer safe sub-solutions.
+As such, current safety algorithms miss data that could be reported as correct, just because they do not
+constrain the solution space enough. A major reason for this lack of progress is that if a problem is NP-hard,
+then its safety version is likely to be hard too. This phenomenon can be found both in classically studied NP-
+hard problems — for example, computing nodes present in all maximum independent sets of an undirected
+graph is NP-hard [21] — as well as in NP-hard problems studied for their application to Bioinformatics, as
+we discuss further in the appendix.
+We introduce our results by focusing on the flow decomposition problem. This is a classical model at the
+core of multi-assembly software for RNA transcripts [33,31,5,50] and viral quasi-species genomes [3,2,44,12],
+and also a standard problem with applications in other fields, such as networking [36,22,13,23] or transporta-
+tion [39,38]. In its most basic optimization form, minimum flow decomposition (MFD), we are given a flow
+in a graph, and we need to decompose it into a minimum number of paths with associated weights, such
+that the superposition of these weighted paths gives the original flow. This is an NP-hard problem, even
+when restricted to DAGs [53,22]. Various approaches have been proposed to tackle the problem, including
+fixed-parameter tractable algorithms [30], approximation algorithms [36,7] and Integer Linear Programming
+formulations [15,46].
+3 The problem AND-Quant from [58] actually handles a more general version of this problem.
+1
+
+In Bioinformatics applications, reads or contigs originating from a mixed sample of genomic sequences
+with different abundances are aligned to a reference. A graph model, such as a splice graph or a variation
+graph, is built from these alignments. Read abundances assigned to the nodes and edges of this graph
+then correspond to a flow in case of perfect data. If this is not the case, the abundance values can either
+be minimally corrected to become a flow, or one can consider variations of the problem where e.g., the
+superposition of the weighted paths is closest (or within a certain range) to the edge abundances [50,5].
+Current safety algorithms for flow decompositions such as [58,27,26,28] compute paths appearing in all
+possible flow decompositions (of any size), even though decompositions of minimum size are assumed to
+better model the RNA assembly problem [30,48,55]. Even dropping the minimality constraint, but adding
+other simple constraints easily renders the problem NP-hard (see e.g., [56]), motivating further study of
+practical safe algorithms for NP-hard problems.
+Contributions. Integer Linear Programming (ILP) is a general and flexible method that has been suc-
+cessfully applied to solve NP-hard problems, including in Bioinformatics. In this paper, we consider graph
+problems whose solution consists of a set of paths (i.e., not repeating nodes) that can be formulated in
+ILP. We introduce a technique that, given an ILP formulation of such a graph problem, can enhance it
+with additional variables and constraints in order to test the safety of a given set of paths. An obvious first
+application of this safety test is to use it with a single path in a straightforward avoid-and-test approach,
+using a standard two-pointer technique that has been used previously to find safe paths for flow decomposi-
+tion. However, we find that a top-down recursive approach that uses the group-testing capability halves the
+number of computationally-intensive ILP calls, resulting in a 3x speedup over the straightforward approach.
+Additionally, we prove that computing all the safe paths for MFDs is an intractable problem, confirming
+the above intuitive claim that if a problem is hard, then also its safety version is hard. We give this proof
+in the appendix by showing that the NP-hardness reduction for MFD by [22] can be modified into a Turing
+reduction from the UNIQUE 3SAT problem.
+On the dataset [47] containing splice graphs from human, zebrafish and mouse transcriptomes, safe
+paths for MFDs (SafeMFD) correctly recover up to 90% of the full RNA transcripts while maintaining a
+99% precision, outperforming, by a wide margin (25% increase), state-of-the-art safety approaches, such as
+extended unitigs [35,24,29], safe paths for constrained path covers of the edges [8], and safe paths for all
+flow decompositions [28,27,26,58]. On the harder dataset by [26], SafeMFD also dominates in a significant
+proportion of splice graphs (built from t ≤ 15 RNA transcripts), recovering more than 95% of the full
+transcripts while maintaining a 98% precision. For larger t, precision drastically drops (91% precision in the
+entire dataset), suggesting that in more complex splice graphs smaller solutions are introduced as an artifact
+of the combinatorial nature of the splice graph, and the minimality condition [30,48,55] is thus incorrect in
+this domain.
+2
+Methods
+2.1
+Preliminaries
+ILP models. In this paper we use ILP models as blackboxes, with as few assumptions as possible to further
+underline the generality of our approach. Let M(V, C) be an ILP model consisting of a set V of variables
+and a set C of constraints on these variables, built from an input graph G = (V, E). We make only two
+assumptions on M. First, that a solution to this model consists of a given number k ≥ 1 of paths P1, . . . , Pk
+in G (in this paper, paths do not repeat vertices). Second, we assume that the k paths are modeled via
+binary edge variables xuvi, for all (u, v) ∈ E and for all i ∈ {1, . . . , k}. More specifically, for all i ∈ {1, . . . , k},
+we require that the edges (u, v) ∈ E for which the corresponding variable xuvi equals 1 induce a path in G.
+For example, if G is a DAG, it is a standard fact (see e.g., [49]) that a path from a given s ∈ V to a given
+2
+
+t ∈ V (an s-t path) can be expressed with the following constraints:
+�
+(u,v)∈E
+xuvi −
+�
+(v,u)∈E
+xvui =
+�
+�
+�
+�
+�
+0,
+if v ∈ V \ {s, t},
+1,
+if v = t,
+−1,
+if v = s.
+(1)
+If G is not a DAG, there are other types of constraints that can be added to the xuvi variables to ensure
+that they induce a path; see, for example, the many formulations in [49]. We will assume that such constraints
+are part of the set C of constraints of M(V, C), but their exact formulation is immaterial for our approach. In
+fact, one could even add additional constraints to C to further restrict the solution space. For example, some
+ILP models from [15,46] handle the case when the input also contains a set of paths (subpath constraints)
+that must appear in at least one of the k solution paths.
+Flow decomposition. In the flow decomposition problem we are given a flow network (V, E, f), where
+G = (V, E) is a (directed) graph with unique source s ∈ V and unique sink t ∈ V , and f assigns a positive
+integer flow value fuv to every edge (u, v) ∈ E. Flow conservation must hold for every node different from s
+and t, namely, the sum of the flow values entering the node must equal the sum of the flow values exiting the
+node. See Figure 1(a) for an example. We say that k s-t paths P1, . . . , Pk, with associated positive integer
+weights w1, . . . , wk, are a flow decomposition (FD) if their superposition equals the flow f. Formally, for
+every (u, v) ∈ E it must hold that
+�
+i∈{1,...,k} s.t.
+(u,v)∈Pi
+wi = fuv.
+(2)
+See Figures 1(b) and 1(c) for two examples. The number k of paths is also called the size of the flow
+decomposition. In the minimum flow decomposition (MFD) problem, we need to find a flow decomposition
+of minimum size.4 On DAGs, a flow decomposition into paths always exists [1], but in general graphs, cycles
+may be necessary to decompose the flow (see e.g. [16] for different possible formulations of the problem).
+For concreteness, we now describe the ILP models from [15] for finding a flow decomposition into k
+weighted paths in a DAG. They consist of (i) modeling the k paths via the xuvi variables (with constraints
+(1)), (ii) adding path-weight variables w1, . . . , wk, and (iii) requiring that these weighted paths form a flow
+decomposition, via the following (non-linear) constraint:
+�
+i∈{1,...,k}
+xuviwi = fuv,
+∀(u, v) ∈ E.
+(3)
+This constraint can then be easily linearized by introducing additional variables and constraints; see e.g. [15]
+for these technical details. However, as mentioned above, the precise formulation of the ILP model M for a
+problem is immaterial for our method. Only the two assumptions on M made above matter for obtaining
+our results.
+Safety. Given a problem on a graph G whose solutions consist of k paths in G, we say that a path P is safe
+if for any solution P1, . . . , Pk to the problem, there exists some i ∈ {1, . . . , k} such that P is a subpath of Pi.
+If the problem is given as an ILP model M, we also say that P is safe for M. We say that P is a maximal
+safe path, if P is a safe path and there is no larger safe path containing P as subpath. [27] characterized safe
+paths for all FDs (not necessarily of minimum size) using the excess flow fP of a path P, defined as the flow
+on the first edge of P minus the flow on the edges out-going from the internal nodes of P, and different from
+the edges of P (see Figure 1(d) for an example). It holds that P is safe for all FDs if and only if fP > 0 [27].
+4 In this paper we work only with integer flow values and weights for simplicity and since this is the most studied
+version of the problem, see e.g., [30]. However, the problem can also be defined with fractional weights [41], and
+in this case the two problems can have different minima on the same input [53]. This fractional case can also be
+modeled by ILP [15], and all the results from our paper also immediately carry over to this variant.
+3
+
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+(a) A flow network with source s and sink t.
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 1
+Figure 3
+xadi = 0
+xeti = 0
+xdei = 0
+xfti = 1
+xsbi = 0
+xbdi = 0
+(b) An MFD into 4 paths of weights 5,3,7,2, respec-
+tively. The green dashed path is a subpath of the
+orange path.
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 1
+Figure 3
+xadi = 0
+xeti = 0
+xdei = 0
+xfti = 1
+xsbi = 0
+xbdi = 0
+(c) An MFD into 4 paths of weights 3,4,1,9, respec-
+tively. The green dashed path is a subpath of the pink
+path.
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+f
+3
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 1
+(d) The two subpaths (red and blue) of the green
+dashed path that are maximal safe paths for all FDs.
+Fig. 1: Flow decompositions and safe paths. The flow network in (a) admits different MFDs, in (b) and in (c).
+The path (s, a, b, c, d) (dashed green) is a maximal safe path for MFDs, i.e., it is a subpath of some path
+of all MFDs and it cannot be extended without losing this property. However, the path (s, a, b, c, d) is not
+safe for all FDs. Indeed, its two subpaths (s, a, b) (dashed red in (d)) and (b, c, d) (dashed blue in (d)) are
+maximal safe paths for all FDs. To see this, note that the excess flow of (s, a, b) is 3, while the excess flow of
+(s, a, b, c) (and of (s, a, b, c, d)) is −6.
+4
+
+The excess flow can be computed in time linear in the length of P (assuming we have pre-computed the flow
+outgoing from every node), giving thus a linear-time verification of whether P is safe.
+A basic property of safe solutions is that any sub-solution of them is also safe. Computing safe paths
+for MFDs can thus potentially lead to joining several safe paths for FDs, obtaining longer paths from the
+unknown sequences we are trying to assemble. See Figure 1 for an example of a maximal safe path for MFDs
+and two maximal subpaths of it that are safe for FDs.
+2.2
+Finding Maximal Safe Paths for MFD via ILP
+We now present a method for finding all maximal safe paths for MFD via ILP. The basic idea is to define an
+inner “safety test” which can be repeatedly called as part of an outer algorithm over the entire instance to
+find all maximal safe paths. Because calls to the ILP solver are expensive, the guiding choice for our overall
+approach is to minimize the number of ILP calls. This inspires us to test the safety of a group of paths as the
+inner safety test, which we achieve by augmenting our ILP model so that it can give us information about
+the safety of the paths in the set. We use this to define a recursive algorithm to fully determine the safety
+status of each path in a group of paths. We can then structure the safety test in either a top-down manner
+(starting with long unsafe paths and shrinking them until they are safe) or a bottom-up manner (starting
+with short safe paths and lengthening them until they become unsafe).
+Safety test (inner algorithm) Let M(V, C) be an ILP model as discussed in Section 2.1; namely, its k
+solution paths are modeled by binary variables xuvi for each (u, v) ∈ E and each i ∈ {1, . . . , k}. We assume
+that M(V, C) is feasible (i.e., the problem admits at least one solution). We first show how to modify the
+ILP model so that, for a given set of paths, it can tell us one of the following: (1) a set of paths that are not
+safe (the remaining being of unknown status), or (2) that all paths are safe. The idea is to maximize the
+number of paths that can be simultaneously avoided from the given set of paths.
+Let P be a set of paths. For each path P ∈ P, we create an auxiliary binary variable γP that indicates:
+γP ≡
+�
+1
+if P was avoided in the solution,
+0
+otherwise.
+(4)
+Since the model solutions are paths (i.e., not repeating nodes), we can encode whether P appears in the
+solution by whether all of the ℓ − 1 edges of P appear simultaneously. Using this fact, we add a new set of
+constraints R(P) that include the γP indicator variables for each path P ∈ P:
+R(P) := {xv1v2i + xv2v3i + · · · + xvℓ−1vℓi
+≤ ℓ − 1 − γP : ∀i ∈ {1, . . . , k}, ∀P ∈ P}.
+(5)
+Next, as the objective function of the ILP model, we require that it should maximize the number of
+avoided paths from P, i,e., the sum of the γP variables:
+max
+�
+P ∈P
+γP .
+(6)
+All paths P such that γP = 1 are unsafe, since they were avoided in some minimum flow decomposition.
+Conversely, if the objective value of Eq. (6) was 0, then γP = 0 for all paths in P, and it must be that all
+paths in P are safe (if not, at least one path could be avoided and increase the objective). We encapsulate
+this group testing ILP in a function GroupTest(M, P) that returns a set N ⊆ P with the properties that:
+(1) if N = ∅, then all paths in P are safe, and (2) if N ̸= ∅, then all paths in N are unsafe (and |N| is
+maximized).
+We employ GroupTest(M, P) to construct a recursive procedure GetSafe(M, P) that determines all safe
+paths in P, as shown in Algorithm 1.
+5
+
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+5
+3
+2
+7
+3
+4
+1
+9
+3
+3
+Figure 1
+Figure 2
+ 
+8
+9
+9
+3
+5
+7
+s
+t
+b
+a
+d
+e
+7
+ c
+10
+10
+f
+3
+3
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai = 1
+xabi = 1
+xbci = 1
+xcdi = 1
+xdfi = 0
+Figure 3
+xadi = 0
+xeti = 1
+xdei = 1
+xfti = 0
+xsbi = 0
+xbdi = 0
+Fig. 2: Illustration of modeling a solution path and a tested path via binary edge variables and safety
+verification constraints. The ith solution path Pi is shown in orange, and a tested path P is shown in dashed
+green. Constraint (5) includes xsai +xabi +xbci +xcdi +xdei ≤ 5−γP . This simplifies to γP ≤ 0, thus forcing
+γP = 0, which indicates P was not avoided in the solution.
+Algorithm 1: Testing a set of paths P for safety.
+Input: A feasible ILP model M(V, C), and a set of paths P
+Output: Those paths P ∈ P that are safe for M(V, C)
+1 Procedure GetSafe(M, P)
+2
+N = GroupTest(M, P) if N = ∅ then
+3
+return P
+4
+else
+5
+return GetSafe(M, P \ N)
+We note that in the special case that |P| = 1, GetSafe(M, P) makes only a single call to the ILP via
+GroupTest(M, P) to determine whether not the given path is safe. With this safety test for a single path, we
+can easily adapt a standard two-pointer approach as the outer algorithm to find all maximal safe paths for
+MFD by starting with some MFD solution P1, . . . , Pk of M(V, C). This same procedure was used in [26] to
+find all maximal safe paths for FD, using an excess flow check as the inner safety algorithm.
+Find all maximal safe paths (outer algorithm) We give two algorithms for finding all maximal safe
+paths. Both algorithms use a similar approach, however the first uses a top-down approach starting from the
+original full solution paths and reports all safe paths (these again must be maximal safe), and then trims all
+the unsafe paths to find new maximal safe paths. The second is bottom-up in that it tries to extend known
+safe subpaths until they cannot be further extended (and at this point must be maximal safe). We present
+the first algorithm in detail and defer discussion of the second to the appendix.
+We say a set of subpaths T = {Pi[li, ri]} is a trimming core provided that for any unreported maximal
+safe path P = Pi[l, r], there is a Pi[li, ri] ∈ T , where li ≤ l ≤ r ≤ ri.
+We will use the original k solution paths {Pi} as our initial trimming core; the complete algorithm is
+given in Algorithm 2. See Fig. 3 in the appendix for an illustration of the algorithm’s initial steps. The
+algorithm first checks if any of the paths in T are safe; if so, these are reported as maximal safe. For those
+paths that were unsafe, it then considers trimming one vertex from the left and one vertex from the right
+to create new subpaths. Of these subpaths, some may be contained in a safe path in T ; these subpaths
+can be ignored as they are not maximal safe. The algorithm recurses on those subpaths whose safety status
+cannot be determined (lines 6–10). In this way, the algorithm maintains the invariant that no paths in T are
+properly contained in a safe path; thus paths reported in line 4 must be maximal safe.
+6
+
+Algorithm 2: An algorithm to compute all maximal safe paths that can be trimmed from a
+trimming core set T .
+Input: An ILP model M and a trimming core set T
+Output: All maximal safe paths for M that are trimmed subpaths of T
+1 Procedure AllMaxSafe-TopDown(M, T )
+2
+S = GetSafe(M, T ) for Pi[li, ri] ∈ S do
+3
+output Pi[li, ri]
+4
+U = T \ S L = {Pi[li + 1, ri] : Pi[li, ri] ∈ U, (ri = |Pi| or Pi[li + 1, ri + 1] ∈ U)}
+R = {Pi[li, ri − 1] : Pi[li, ri] ∈ U, (li = 1 or Pi[li − 1, ri − 1] ∈ U)} P = L ∪ R if P ̸= ∅ then
+5
+AllMaxSafe-TopDown(M, P)
+3
+Experiments
+To test the performance of our methods, we computed safe paths using different safety approaches and re-
+ported the quality and running time performances as described below. Additional details on the experimental
+setup are given in the appendix.
+Implementation details – SafeMFD. We implemented the previously described algorithms to compute
+all maximal safe paths for minimum flow decompositions in Python. The implementation, SafeMFD, uses
+the package NetworkX [20] for graph processing and the package gurobipy [19] to model and solve the ILPs
+and it is openly available5. Our fastest variant (see Table 2 in the appendix for a comparison of running
+times) implements Algorithm 2 using the group testing in Algorithm 1. We used this variant to compare
+against other safety approaches. All tested variants of SafeMFD implement the following two optimizations:
+1.
+Before processing an input flow graph we contract it using Y-to-V contraction [51], which is known [30]
+to maintain (M)FD solution paths. Moreover, since edges in the contracted graph correspond to extended
+unitigs [35,24,29], source-to-sink edges are further removed from the contracted graph and reported as
+safe. As such, our algorithms compute all maximal safe paths for funnels [17,26] without using the ILP.
+2.
+Before testing the safety of a path we check if its excess-flow [26] is positive. If this is the case, the
+path is removed from the corresponding test. Having positive excess flow implies safety for all flow
+decomposition and thus also safety for minimum flow decompositions.
+Safety approaches tested. We compare the following state-of-the-art safety approaches:
+EUnitigs:
+Maximal paths made up of a prefix of nodes with in-degree one followed by nodes with out-
+degree one; also called extended unitigs [51,35,24,29]. We use the C++ implementation provided by Khan
+et al. [26] (which computes only the extended unitigs contained in FD paths).
+SafeFlow:
+Maximal safe paths for all flow decompositions [26]. We use the C++ implementation provided
+by Khan et al. [26].
+SafeMFD:
+Maximal safe paths for all minimum flow decompositions, as proposed in this work. Every
+flow graph processed is given a time budget of 2 minutes. If a flow graph consumes its time budget, the
+solution of SafeFlow is output instead.
+SafeEPC:
+Maximal safe paths for all constrained path covers of edges. Previous authors [8,26] have con-
+sidered safe path covers of the nodes, but for a more fair comparison, we instead use path covers of edges.
+To this end, we transform the input graphs by splitting every edge by adding a node in the middle and
+run the C++ implementation provided by the authors of [8]. Since flow decompositions are path covers
+of edges, safe paths for all edge path covers are subpaths of safe paths for MFD. However, we restrict
+the path covers to those of minimum size and minimum size plus one, as recommended by the authors
+of [8] to obtain good coverage results while maintaining high precision.
+5 https://github.com/algbio/mfd-safety
+7
+
+All safety approaches require a post processing step for removing duplicates, prefixes and suffixes. We
+use the C++ implementation provided by [26] for this purpose.
+Datasets. We use two datasets of flow graphs inspired by RNA transcript assembly. The datasets were
+created by simulating abundances on a set of transcripts and then perfectly superposing them into a splice
+graphs that are guaranteed to respect flow conservation. As such, the ground truth corresponds to a flow
+decomposition (not necessarily minimum). To avoid a skewed picture of our results we filtered out trivial
+instances with a unique flow decomposition (or funnels, see [17,26]) from the two datasets.6
+Catfish:
+Created by [48], it includes 100 simulated human, mouse and zebrafish transcriptomes using Flux-
+Simulator [18] as well as 1,000 experiments from the Sequence Read Archive simulating abundances using
+Salmon [40]. We took one experiment per dataset, which corresponds to 27,696 non-trivial flow graphs.
+RefSim:
+Created by [8] from the Ensembl [57] annotated transcripts of GRCh.104 homo sapiens reference
+genome, and later augmented by Khan et al. [26] with simulated abundances using the RNASeqRead-
+Simulator [32]. This dataset has 10,323 non-trivial graphs.
+Quality metrics. We use the same quality metrics employed by previous multi-assembly safety approaches [8,26].
+We provide a high-level description of them for completeness.
+Weighted precision of reported paths:
+As opposed to normal precision, the weighted version considers
+the length of the reported subpaths. It is computed as the total length of the correctly reported subpaths
+divided by the total length of all reported subpaths. A reported subpath is considered correct if and only
+if it is a subpath of some path in the ground truth (exact alignment of exons/nodes).
+Maximum coverage of a ground truth path P:
+The longest segment of P covered by some reported
+subpath (exact alignment of exons/nodes), divided by |P|.
+We compute the weighted precision of a graph as the average weighted precision over all reported
+paths in the graph, and the maximum coverage of a graph as the average maximum coverage over all
+ground truth paths in the graph.
+F-Score of a graph:
+Harmonic mean between weighted precision and maximum coverage of a graph,
+which assigns a global score to the corresponding approach on the graph.
+These metrics are computed per flow graph and reported as an average. In the case of the Catfish dataset
+the metrics are computed in terms of exons (nodes), since genomic coordinates of exons are missing, whereas
+in the case of the RefSim dataset the metrics are computed in terms of genomic positions, as this information
+is present in the input.
+4
+Results and Discussion
+In the Catfish dataset, EUnitigs and SafeFlow ran in less than a second, while SafeEPC took approximately
+30 seconds to compute. On the other hand, solving a harder problem, SafeMFD took approximately 1.5
+hours to compute in the rest of the dataset, timing out in only 54 graphs (we use a cutoff of 2 minutes), i.e.,
+only 0.2% of the entire dataset. This equates to only 0.2 seconds on average per solved graph, underlying
+the scalability of our approach.
+Table 1 shows that SafeMFD, on average, covers close to 90% of the ground truth paths, while maintaining
+a high precision (99%). This corresponds to an increase of approximately 25% in coverage against its closest
+competitor SafeFlow. SafeMFD also dominates in the combined metric of F-Score, being the only safe
+approach with F-Score over 90%. Figure 4 in the appendix shows the metrics on graphs grouped by number
+t of ground truth paths, indicating the dominance in coverage and F-Score of SafeMFD across all values of
+t, and indicating that the decrease in precision appears for large values of t (t ≥ 12).
+6 The exact datasets used in our experiments can be found at https://zenodo.org/record/7182096.
+8
+
+Table 1: Summary of quality metrics for both datasets. For Catfish, the metrics are computed in terms of
+nodes/exons and for RefSim in terms of genomic positions; t is the number of ground truth paths.
+Dataset
+Graphs
+Algorithm Max. Coverage Wt. Precision F-Score
+Catfish
+All
+(100%)
+EUnitigs
+0.60
+1.00
+0.74
+SafeEPC
+0.60
+0.99
+0.74
+SafeFlow
+0.71
+1.00
+0.82
+SafeMFD
+0.88
+0.99
+0.93
+RefSim
+t ≤ 10
+(68%)
+EUnitigs
+0.72
+1.00
+0.83
+SafeEPC
+0.73
+1.00
+0.84
+SafeFlow
+0.84
+1.00
+0.91
+SafeMFD
+0.97
+0.99
+0.98
+t ≤ 15
+(84%)
+EUnitigs
+0.70
+1.00
+0.82
+SafeEPC
+0.71
+1.00
+0.83
+SafeFlow
+0.83
+1.00
+0.90
+SafeMFD
+0.96
+0.98
+0.97
+All
+(100%)
+EUnitigs
+0.68
+1.00
+0.80
+SafeEPC
+0.69
+0.99
+0.81
+SafeFlow
+0.81
+1.00
+0.89
+SafeMFD
+0.93
+0.91
+0.90
+In the harder RefSim dataset, EUnitigs and SafeFlow also ran in less than a second, while SafeEPC
+took approximately 2 minutes. In this case, SafeMFD ran out of time in 1,562 graphs (15% of the entire
+dataset); however, recall that in these experiments we allow a time budget of only 2 minutes. In the rest of
+the dataset, it took approximately 7.5 hours in total, corresponding to only 3 seconds on average per graph,
+again underlying that our method, even though it solves many NP-hard problems for each input graph,
+overall scales sufficiently well.
+Table 1 shows that again SafeMFD dominates in coverage, being the only approach obtaining coverage
+over 90%, with is a 15% improvement over SafeFlow. This time its precision drops to close to 90%, and
+obtaining an F-Score of 90%, very similar to its closest competitor, SafeFlow. However, recall that coverage
+is computed only from correctly aligned paths, thus the drop in precision comes only from safe paths not
+counting in the coverage metric. If we restrict the metrics to graphs with at most 15 ground truth paths,
+which is still a significant proportion (84%) of the entire dataset, then SafeMFD has a very high precision
+(98%) while improving coverage by 15% with respect to SafeFlow. Thus, the drop in precision occurs in
+graphs with a large number of ground truth paths, which can also be corroborated by Figure 5 in the
+appendix.
+These drops in precision (both in RefSim and Catfish) for large t can be explained by the fact that a
+larger number of ground truth paths produces more complex splice graphs and introduces more artificial
+solutions of potentially smaller size. As such, the larger t, the less likely that the ground truth is a minimum
+flow decomposition of the graph, and thus the more likely that SafeMFD reports incorrect solutions. This
+motivates future work on safety not only on minimum flow decompositions but also in flow decompositions
+of at most a certain size, analogously to how it is done for SafeEPC. This is still easily achievable with
+our framework by just changing the ILP blackbox, and keeping everything else unchanged (e.g., the inner
+and outer algorithms). Namely, instead of formulating the ILP model M(V, C) to admit solutions of exactly
+optimal k paths, it can be changed to allow solutions of at most some k′ paths, with k′ greater than the
+optimal k. If k′ is also greater than the number of ground truth paths in these complex graphs, then safe
+paths are fully correct, meaning that we overall increase precision.
+9
+
+5
+Conclusion
+RNA assembly is a difficult problem in practice, with even the top tools reporting low precision values. While
+there are still many issues that can introduce uncertainty in practice, we can now provide a major source
+of additional information during the process: which RNA fragments must be included in any parsimonious
+explanation of the data? Though others have considered RNA assembly in the safety framework [58,26], we
+are the first to show that safety can be practically used even when we look for optimal (i.e., minimum) size
+solutions. Our experimental results show that safe paths for MFD clearly outperform other safe approaches
+for the Catfish dataset, commonly used in this field. On a significant proportion of the second dataset, safe
+paths for MFD still significantly outperforms other safe methods.
+More generally, this is the first work to show that the safety framework can be practically applied to
+NP-hard problems, where the inner algorithm is an efficient test of safety of a group of paths, and the outer
+algorithm guides the applications of this test. Because our method was very successful on our test data set,
+there is strong motivation to try the approach to on other NP-hard graph problems whose solutions are
+sets of paths. For example, we could study other variations on MFD, such as finding flow decompositions
+minimizing the longest path (NP-hard when flow values are integer [4,43]). The approach given in this paper
+can also be directly extended to find decompositions into both cycles and paths [16], though not trails and
+walks, because they repeat edges. We could also formulate a safety test for classic NP-hard graph problems
+like Hamiltonian path.
+Acknowledgements This work was partially funded by the European Research Council (ERC) under
+the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 851093,
+SAFEBIO), partially by the Academy of Finland (grants No. 322595, 352821, 346968), and partially by
+the US National Science Foundation (NSF) (grants No. 1759522, 1920954).
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+
+A
+Additional Figures
+P1
+P2
+P3
+P4
+P1
+P2
+P3
+P4
+a) Group Testing - Initial Iteration
+c) Group Testing -  Second Iteration
+b) Identifying Safe Paths
+P2
+P2
+P4
+P4
+(a) First group test
+P1
+P2
+P3
+P4
+P1
+P2
+P3
+P4
+a) Group Testing - Initial Iteration
+c) Group Testing -  Second Iteration
+b) Identifying Safe Paths
+P2
+P2
+P4
+P4
+(b) Result: {P1, P3} are safe, {P2, P4}
+are unsafe
+P1
+P2
+P3
+P4
+P1
+P2
+P3
+P4
+a) Group Testing - Initial Iteration
+c) Group Testing -  Second Iteration
+b) Identifying Safe Paths
+P2
+P2
+P4
+P4
+(c) Second group test
+Fig. 3: Illustration of the initial group tests performed by Algorithm 2. Fig. 3(a) shows the first group test
+(using Algorithm 1) on MFD solution paths {P1, P2, P3, P4}; suppose {P1, P3} were safe (Fig. 3(b)); these
+are then reported as maximal safe. In this case we trim {P2, P4} on both the left and right and make the
+next group test shown in Fig. 3(c).
+13
+
+(a) Weighted Precision
+(b) Maximum Coverage
+(c) F-Score
+Fig. 4: Quality metrics on graphs distributed by number of paths in the ground truth for the Catfish dataset.
+The metrics are computed in terms of exons/nodes.
+(a) Weighted Precision
+(b) Maximum Coverage
+(c) F-Score
+Fig. 5: Quality metrics on graphs distributed by number of paths in the ground truth for the RefSim dataset.
+The metrics are computed in terms of genomic positions.
+B
+Additional Algorithms and Experimental Results
+B.1
+The bottom-up algorithm
+Algorithm 3, detailed below, uses a bottom-up group-testing strategy to find all maximal safe paths.
+Definition 1. We say a set of subpaths E = {Pi[li, ri]} is an extending core provided all paths in E are safe
+and for any unreported maximal safe path P = Pi[l, r], there is a Pi[li, ri] ∈ E, where l ≤ li ≤ ri ≤ r.
+Note that maximal FD-safe subpaths provide an extending core (as well just the set of all single-edge
+subpaths in each path). Algorithm 3 provides an algorithm to find all maximal safe paths based on group
+testing, starting from an extending core. The idea is to try both left-extending (by one) and right-extending
+(by one) each subpath in the core; if neither of these extensions are safe, then we know that that core subpath
+must be maximal safe. Testing all extensions can done quickly using Algorithm 1. We then recurse on a new
+core set consisting of those extensions that were found to be safe.
+14
+
+1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+SafeMFD
+0.0
+3
+4
+5
+6
+7
+8
+9
+10 11 12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+1011.12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+1011.12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+SafeMFD
+0.0
+3
+4
+5
+6
+7
+8
+9
+10 11 12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+10.11.12
+13
+14
+15
+# Ground truth paths1.0
+0.8
+0.6
+0.4
+0.2
+EUnitigs
+SafeEPC
+SafeFlow
+0.0
+SafeMFD
+3
+4
+5
+6
+7
+8
+9
+1011.12
+13
+14
+15
+# Ground truth pathsAlgorithm 3: An algorithm to output all maximal safe subpaths that can be extended from an
+extending core set E.
+Input: An ILP model M and an extending core set E
+Output: All maximal safe paths for M that extend some path from E
+1 Procedure AllMaxSafe-BottomUp(M, E)
+2
+L = {Pi[li − 1, ri] : Pi[li, ri] ∈ E, li > 1} R = {Pi[li, ri + 1] : Pi[li, ri] ∈ E, ri < |Pi|} P = L ∪ R S
+= GetSafe(M, P) for Pi[li, ri] ∈ E do
+3
+if Pi[li − 1, ri] /∈ S and Pi[li, ri + 1] /∈ S then
+4
+output Pi[li, ri]
+5
+if S ̸= ∅ then
+6
+AllMaxSafe-BottomUp(M, S)
+B.2
+The two-pointer algorithm
+As we observed in Section 2.2, we can test whether a single path P is safe using one ILP call. We will
+assume that this test is encapsulated as a procedure IsSafe(M, P). Once we can test whether a single path
+is safe for M(V, C), we can adopt a standard approach to compute all maximal safe paths. Namely, we start
+by computing one solution of M(V, C), P1, . . . , Pk and then compute maximal safe paths by a two-pointer
+technique that for each path Pi, finds all maximal safe paths by just a linear number of calls to the procedure
+IsSafe [26].
+This works as follows. We use two pointers, a left pointer L, and a right pointer R. Initially, L points to
+the first node of path Pi and R to the second node. As long as the subpath of Pi between L and R is safe,
+we move the right pointer to the next node on Pi. When this subpath is not safe, we output the subpath
+between L and the previous location of R as a maximal safe path, and we start moving the left pointer to
+the next node on Pi, until the subpath between L and R is safe. We stop the procedure once we reach the
+end of Pi. We summarize this procedure as Algorithm 4; see also Figure 6 for an example.
+ 
+1
+8
+3
+2
+5
+6
+ 4
+7
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai + xabi + xbci + xcdi + xdfi ≤ 4
+xsai + xabi + xbci + xcdi ≤ 3
+xabi + xbci + xcdi + xdfi ≤ 3
+R
+L
+R
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+L
+L
+R
+(a) Current iteration
+ 
+1
+8
+3
+2
+5
+6
+ 4
+7
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai + xabi + xbci + xcdi + xdfi ≤ 4
+xsai + xabi + xbci + xcdi ≤ 3
+xabi + xbci + xcdi + xdfi ≤ 3
+R
+L
+R
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+L
+L
+R
+(b) Right pointer movement
+ 
+1
+8
+3
+2
+5
+6
+ 4
+7
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+ 
+s
+t
+b
+a
+d
+e
+ c
+f
+xsai + xabi + xbci + xcdi + xdfi ≤ 4
+xsai + xabi + xbci + xcdi ≤ 3
+xabi + xbci + xcdi + xdfi ≤ 3
+R
+L
+R
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+∀i ∈ {1,…, k}
+L
+L
+R
+(c) Left pointer movement
+Fig. 6: Illustration of the two-pointer algorithm applied on a flow decomposition path Pi (in orange). In each
+sub-figure, the subpath P (dashed green) between the nodes pointed by the left pointer L and the right
+pointer R is tested for safety, by adding constraints S(P). In (a), IsSafe(M, P) returns True, and the right
+pointer advances on Pi. In (b), IsSafe(M, P) returns False, and the previous subpath from (a) is output as
+a maximal safe path. In (c), the left pointer has advanced, and the new path P is tested for safety.
+15
+
+Algorithm 4: The two-pointer algorithm applied to compute all maximal subpaths of a given
+solution path Pi
+Input: An ILP model M and one of its k solution paths, Pi = (v1, . . . , vt), t ≥ 2
+Output: All maximal safe subpaths of Pi for M
+1 Procedure AllMaxSafe-TwoPointer(M, Pi)
+2
+L ← 1, R ← 2 while True do
+3
+while IsSafe(M, Pi[L, R]) and R ≤ t do
+4
+R ← R + 1
+5
+output Pi[L, R − 1] if R > t then return;
+6
+while not IsSafe(M, Pi[L, R]) do
+7
+L ← L + 1
+Dataset (# Graphs)
+Variant
+Time (hh:mm:ss) # ILP calls
+Catfish
+(27,613)
+TopDown
+01:13:27
+124,676
+BottomUp
+03:22:13
+212,774
+TwoPointer
+04:21:44
+226,365
+TwoPointerBin
+03:31:57
+216,540
+RefSim
+(5,808)
+TopDown
+04:38:41
+55,450
+BottomUp
+11:55:20
+76,837
+TwoPointer
+13:48:00
+127,352
+TwoPointerBin
+11:34:02
+119,218
+Table 2: Running times and number of ILP calls in four different variants of SafeMFD.
+B.3
+Running time experiments among different variants proposed
+We conducted the experiments on an isolated Linux server with AMD Ryzen Threadripper PRO 3975WX
+CPU with 32 cores (64 virtual) and 504GB of RAM. Time and peak memory usage of each program were
+measured with the GNU time command. SafeMFD was allowed to run Gurobi with 12 threads. All C++
+implementations were compiled with optimization level 3 (-O3 flag). Running time and peak memory is
+computed and reported per dataset.
+SafeMFD includes the following four variants computing maximal safe paths:
+TopDown : Implements Algorithm 2 using the group testing in Algorithm 1.
+BottomUp : Implements Algorithm 3 (Appendix B.1) using the group testing in Algorithm 1.
+TwoPointer : Implements Algorithm 4 (Appendix B.2), the traditional two-pointer algorithm [26].
+TwoPointerBin : Same as previous variant, but it additionally replaces the linear scan employed to extend
+and reduce the currently processed safe path by a binary search7.
+To compare between our four different variants we first run them all on every dataset, and then filter
+out those graphs that ran out of time in some variant. This way we ensure that no variant consumes its
+time budget and thus our running time measurements are not skewed by the unsuccessful inputs’ timeouts.
+Applying this filter we removed 83 graphs from the Catfish dataset (0.3%) and 4,515 graphs from the RefSim
+dataset (43.74%).
+Table 2 shows the running times and number of ILP calls of the different variants on both datasets.
+TopDown clearly outperforms the rest, being at least twice as fast, and performing (roughly) half many ILP
+calls. While BottomUp is analogous to TopDown, the superiority of the latter can be explained by the length
+maximal safe paths. Since maximal safe paths are long it is faster to obtain them by reducing unsafe paths
+7 The binary search is only applied if the search space is larger than a constant threshold set experimentally.
+16
+
+(TopDown) than by extending safe paths (BottomUp and both TwoPointer variants). On the other hand,
+TwoPointer is the slowest variant and BottomUp and TwoPointerBin obtain similar improvements (over
+TwoPointer) by following different strategies. While BottomUp reduces the number of ILP calls more than
+TwoPointerBin (better appreciated in the RefSim dataset), the ILP calls of BottomUp take longer (since
+BottomUp tests several paths at the same time and TwoPointerBin only one), and thus the total running
+times of both is similar. This motivates future work on combining both approaches, while processing the
+paths starting from unsafe (as in TopDown) for better performance.
+C
+Hardness of Testing MFD Safety
+In this section we give a Turing-reduction from the UNIQUE 3SAT problem (U3SAT) to the problem of
+determining if a given path P in a flow network G is safe for minimum flow decomposition (call this problem
+MFD-SAFETY ). A 3SAT instance belongs to U3SAT if and only if it has exactly one satisfying assignment.
+U3SAT has been shown to be NP-hard under randomized reductions [52], but it is open as to whether it is
+NP-hard in general.
+The reduction leverages the construction in [22] that reduces 3SAT to minimum flow decomposition. We
+first briefly review this construction: A variable gadget (see Fig. 4 in [22]) is created for each 3SAT variable x
+and a clause gadget (see Fig. 5 in [22]) is created for each 3SAT clause. Positive literals in each clause receive
+flow from the left side of the corresponding variable gadget, whereas negative literals receive flow from the
+right side. Theorem VI.1 in [22] establishes that a 3SAT instance is satisfiable if and only if the constructed
+flow network has a minimum flow decomposition of a certain size. Any flow decomposition achieving this
+size must have a specific structure; in particular, there must be a flow path of weight 4 that either travels
+up the left side of the gadget (setting x to TRUE), or the right side (setting x to FALSE).
+⋯
+4
+4
+4
+4
+4
+4
+⋯
+4
+4
+4
+4
+t(x)
+s(x)
+Fig. 7: The variable gadget from [22], showing only the weight 4 edges (other edges have weights from
+{1, 2}). A key property established in [22] is that if the 3SAT instance is satisfiable then in a minimum flow
+decomposition, a weight 4 flow path must travel up either the left side of the gadget (as shown), or the
+right side. A left flow path indicates the variable should be set to TRUE, while right indicates FALSE. We
+leverage this construction to reduce U3SAT to MFD-SAFETY.
+Theorem 1. There is a polynomial time Turing-reduction from U3SAT to MFD-SAFETY.
+17
+
+Proof. To obtain the desired Turing-reduction algorithm, instead of checking the size of the MFD, we will
+instead sequentially check the MFD-SAFETY of the aforementioned side paths traveling up the left and
+right sides of each variable gadget. Provided each variable gadget has exactly one safe side path we can then
+check the corresponding truth assignment to see if each clause is satisfied. If yes, we accept the instance as
+belonging to U3SAT, otherwise we reject.
+Suppose the instance does belong to U3SAT. In this case there is a satisfying assignment so the MFD
+must have the structure as described above. Furthermore, since there is exactly one satisfying assignment,
+exactly one side path of each variable gadget must be safe and so our algorithm finds it and then verifies that
+the truth assignment satisfies each clause, thus accepting the instance. On the other hand, if the instance does
+not belong to U3SAT it could either be unsatisfiable or have multiple satisfying assignments. If unsatisfiable,
+no matter whether the safety checks pass, the corresponding assignment will not satisfy all clauses, so the
+instance will be rejected. If there are multiple solutions, then any variable that can be both TRUE and
+FALSE will not have a safe side path in the MFD. This means the safety check will fail and the instance
+will again be rejected.
+⊓⊔
+18
+

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+arXiv:2301.00278v1  [math.CO]  31 Dec 2022
+Isometric path antichain covers: beyond hyperbolic graphs∗
+Dibyayan Chakraborty†
+Florent Foucaud‡
+January 3, 2023
+Abstract
+The isometric path antichain cover number of a graph G, denoted by ipacc (G), is a graph pa-
+rameter that was recently introduced to provide a constant factor approximation algorithm for Iso-
+metric Path Cover, whose objective is to cover all vertices of a graph with a minimum number
+of isometric paths (i.e. shortest paths between their end-vertices). This parameter was previously
+shown to be bounded for chordal graphs and, more generally, for graphs of bounded chordality and
+bounded treelength. In this paper, we show that the isometric path antichain cover number remains
+bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta,
+prism, pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively studied in
+Metric Graph Theory. The class of (theta, prism, pyramid)-free graphs are extensively studied in
+Structural Graph Theory, e.g. in the context of the Strong Perfect Graph Theorem. The class of
+outerstring graphs is studied in Geometric Graph Theory and Computational Geometry. Our results
+imply a constant factor approximation algorithm for Isometric Path Cover on all the above graph
+classes. Our results also show that the distance functions of these (structurally) different graph classes
+are more similar than previously thought.
+1
+Introduction
+A path is isometric if it is a shortest path between its endpoints. An isometric path cover of a graph G
+is a set of isometric paths such that each vertex of G belongs to at least one of the paths. The isometric
+path number of G is the smallest size of an isometric path cover of G. Given a graph G and an integer k,
+the objective of the algorithmic problem Isometric Path Cover is to decide if there exists an isometric
+path cover of cardinality at most k. Isometric Path Cover has been introduced and studied in the
+context of pursuit-evasion games [1, 2] and used in the context of Product Structure Theorems [15].
+The goal of this paper is to continue the study of approximation algorithms for Isometric Path
+Cover on several graph classes.
+We do so by continuing the study of a recently introduced graph
+parameter which seems interesting in its own right, as it encapsulates several previously unrelated graph
+classes.
+Isometric Path Cover has also been studied from a structural point of view: the cardinalities
+of the optimal solution have been determined for square grids [17], hypercubes [18], complete r-partite
+graphs [24] and Cartesian products of complete graphs [24], and it was recently proved that the pathwidth
+of a graph is always upper-bounded by the size of its smallest isometric path cover [16]. However, until
+recently the algorithmic aspects of Isometric Path Cover remained unexplored. The problem is easy
+to solve on trees and more generally, on block graphs [23] but remains hard on chordal graph, i.e. graphs
+without any induced cycle of length at least 4 [7]. It can be approximated in polynomial time within a
+factor of log(d) for graphs of diameter d by a greedy algorithm [27] and solved in polynomial time for every
+∗This
+research
+was
+partially
+financed
+by
+the
+IFCAM
+project
+“Applications
+of
+graph
+homomorphisms”
+(MA/IFCAM/18/39), the ANR project GRALMECO (ANR-21-CE48-0004) and the French government IDEX-ISITE ini-
+tiative 16-IDEX-0001 (CAP 20-25).
+†Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
+‡Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-
+Ferrand, France
+1
+
+fixed value of k by an XP algorithm [16]. In a quest to find constant factor approximation algorithms
+for Isometric Path Cover, Chakraborty et al. [7] introduced a parameter called the isometric path
+antichain cover number of graphs, denoted by ipacc (G) (see Section 2 for a definition) and proved a
+result directly implying the following (see [7, Proposition 10]).
+Proposition 1 ([7]). For a graph G, if ipacc (G) ≤ c, then Isometric Path Cover admits a polynomial-
+time c-approximation algorithm on G.
+Proposition 1 is proved by a simple approximation algorithm described as follows. For each vertex r
+of the graph, perform a Breadth-First Search at this vertex. Remove edges joining any vertices at the
+same distance from r, and orient all edges towards r. The resulting directed acyclic graph can be seen as
+the Hasse diagram of a poset. Compute a chain covering of that poset using classic methods related to
+Dilworth’s theorem. The chains are the isometric paths of the solution. Keep the smallest of all solutions
+over all choices of r.
+Using Proposition 1, the above algorithm was shown to be a constant factor approximation algo-
+rithm for many graph classes, including interval graphs, chordal graphs, and more generally, graphs with
+bounded treelength. Indeed, on all these graph classes, the isometric path antichain cover number is
+shown to be bounded by a constant (note that one does not need to compute this parameter for the
+algorithm to function: it serves only in the analysis of the approximation ratio of the algorithm). As
+noted in [7], this parameter may be unbounded on general graphs, for example for the class of hypercubes
+or square grids.
+In this paper, we continue to study the boundedness of the isometric path antichain cover number
+of various graph classes. Specifically, we consider three structurally unrelated graph classes, namely,
+hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs, which extends the above
+work to strictly larger graph classes.
+Hyperbolic graphs: A graph G is said to be δ-hyperbolic [19] if for any four vertices u, v, x, y, the two
+larger of the three distance sums d (u, v)+d (x, y), d (u, x)+d (v, y) and d (u, y)+d (v, x) differ by at most
+2δ. A graph class G is hyperbolic if there exists a constant δ such that every graph G ∈ G is δ-hyperbolic.
+This parameter was first introduced by Gromov in the context of automatic groups [19] in relation with
+their Cayley graphs. The hyperbolicity of a tree is 0, and in general, hyperbolicity seems to measure
+how much the distance function of a graph deviates from a tree metric. Many structurally defined graph
+classes like chordal graphs, cocomparability graphs, asteroidal-triple free graphs, graphs with bounded
+chordality or treelength are hyperbolic graphs [8, 21]. Moreover, hyperbolicity has been found to capture
+important properties of several large practical graphs such as the Internet [26] or database relations [31].
+Due to its importance in discrete mathematics, algorithms, metric graph theory, researchers have studied
+various algorithmic aspects of hyperbolic graphs [8, 12, 9, 13]. Note that graphs with diameter 2 are
+hyperbolic, which may contain any graph as an induced subgraph.
+(theta, prism, pyramid)-free graphs: A theta is a graph made of three vertex-disjoint induced paths
+P1 = a . . . b, P2 = a . . . b, P3 = a . . . b of lengths at least 2, and such that no edges exist between the paths
+except the three edges incident to a and the three edges incident to b. See Figure 2 for an illustration. A
+pyramid is a graph made of three induced paths P1 = a . . . b1, P2 = a . . . b2, P3 = a . . . b3, two of which
+have lengths at least 2, vertex-disjoint except at a, and such that b1b2b3 is a triangle and no edges exist
+between the paths except those of the triangle and the three edges incident to a. A prism is a graph
+made of three vertex-disjoint induced paths P1 = a1 . . . b1, P2 = a2 . . . b2, P3 = a3 . . . b3 of lengths at
+least 1, such that a1a2a3 and b1b2b3 are triangles and no edges exist between the paths except those of
+the two triangles. A graph G is (theta, pyramid, prism)-free if G does not contain any induced subgraph
+isomorphic to a theta, pyramid or prism. A graph is a 3-path configuration if it is a theta, pyramid or
+prism. The study of 3-path configurations dates back to the works of Watkins and Meisner [32] in 1967
+and plays “special roles” in the proof of the celebrated Strong Perfect Graph Theorem [10, 14, 28, 30].
+Important graph classes like chordal graphs, circular arc graphs, universally-signable graphs [11] exclude
+all 3-path configurations. Popular graph classes like perfect graphs, even hole-free graphs exclude some
+2
+
+Bounded isometric path
+antichain cover number
+bounded hyperbolicity *
+(t-theta, t-prism, t-pyramid)-
+free *
+Outerstring *
+circle *
+(theta,prism,pyramid)-
+free *
+Universally signable *
+bounded tree-length
+bounded chordality
+bounded diameter
+chordal
+AT-free
+Interval
+circular arc *
+Permutation
+Figure 1: Inclusion diagram for graph classes discussed here (and related ones). If a class A has an
+upward path to class B, then A is included in B. For graphs in the gray classes, the complexity of
+Isometric Path Cover is open; for all other graph classes, it is NP-complete. For all shown graph
+classes, Isometric Path Cover is constant-factor approximable in polynomial time. Constant factor
+approximation algorithms for Isometric Path Cover on graph classes marked with * are contributions
+of this paper.
+b
+a
+b1
+b2
+b3
+a
+b1
+b2
+b3
+a1
+a2
+a3
+(a)
+(b)
+(c)
+(d)
+Figure 2: (a) Theta, (b) Pyramid, (c) Prism, (d) Outerstrings. The figure shows that the graph K2,3,
+which is also a theta, is an outerstring graph.
+of the 3-path configurations. Note that, (theta, prism, pyramid)-free graphs are not hyperbolic. To see
+this, consider a cycle C of order n. Clearly, C excludes all 3-path configurations and has hyperbolicity
+Ω(n).
+Outerstring graphs: A set S of simple curves on the plane is grounded if there exists a horizontal line
+containing one endpoint of each of the curves in S. A graph G is an outerstring graph if there is a collection
+C of grounded simple curves and a bijection between V (G) and C such that two curves in S if and only
+if the corresponding vertices are adjacent in G. See Figure 2(d) for an illustration. The term “outerstring
+graph” was first used in the early 90’s [22] in the context of studying intersection graphs of simple curves
+on the plane. Many well-known graph classes like chordal graphs, circular arc graphs, circle graphs
+(intersection graphs of chords of a circle), or cocomparability graphs are also outerstring graphs and
+thus, motivated researchers from the geometric graph theory and computational geometry communities
+to study algorithmic and structural aspects of outerstring graphs and its subclasses [4, 5, 6, 20, 25]. Note
+that, in general, outerstring graphs may contain a prism, pyramid or theta as an induced subgraph.
+Moreover, cycles of arbitrary order are outerstring graphs, implying that outerstring graphs are not
+hyperbolic.
+It is clear from the above discussion that the classes of hyperbolic graphs, (theta, prism, pyramid)-free
+3
+
+graphs, and outerstring graphs are pairwise incomparable (with respect to the containment relationship).
+1.1
+Our contributions
+The main contribution of this paper is to show that the isometric path antichain cover number (see
+Section 2 for a definition) remains bounded on hyperbolic graphs, (theta, pyramid, prism)-free graphs,
+and outerstring graphs. Specifically, we prove the following theorems.
+Theorem 2. Let G be a graph with hyperbolicity δ. Then, ipacc (G) ≤ 12δ + 6.
+Theorem 3. Let G be a (theta, pyramid, prism)-free graph. Then, ipacc (G) ≤ 71.
+Theorem 4. Let G be an outerstring graph. Then, ipacc (G) ≤ 95.
+To the best of our knowledge, the isometric path antichain cover number being bounded (by con-
+stant(s)) is the only known non-trivial property shared by any two or all three of these graph classes.
+To provide a unified proof of Theorem 3 and 4, we study a more general graph class called (t-theta,
+t-pyramid, t-prism)-free graphs [29] (see Section 4 for definition). When t = 1, (t-theta, t-pyramid, t-
+prism)-free graphs are exactly (theta, prism, pyramid)-free graphs. Moreover, we show that all outerstring
+graphs are (4-theta, 4-pyramid, 4-prism)-free graphs (Lemma 16). We prove the following.
+Theorem 5. For t ≥ 1, let G be a (t-theta, t-pyramid, t-prism)-free graph. Then ipacc (G) ≤ 8t + 63.
+Due to Proposition 1 and the above theorems, we also have the following corollary.
+Corollary 6. There is an approximation algorithm for Isometric Path Cover with approximation
+ratio
+(a) 12δ + 6 on δ-hyperbolic graphs,
+(b) 73 on (theta, prism, pyramid)-free graphs, and
+(c) 95 on outerstring graphs.
+(d) 8t + 63 on (t-theta, t-pyramid, t-prism)-free graphs.
+Organisation: In Section 2, we introduce the recall some definitions and some results. In Section 3 we
+prove Theorem 2. In Section 4, we prove Theorems 3 and 5. In Section 5, we prove Theorem 4. We
+conclude in Section 6.
+2
+Definitions and preliminary observations
+In this section, we formally recall the definition of isometric path antichain cover number of graphs from
+[7] and some related observations. A sequence of distinct vertices forms a path P if any two consecutive
+vertices are adjacent. Whenever we fix a path P of G, we shall refer to the subgraph formed by the
+edges between the consecutive vertices of P. The length of a path P, denoted by |P|, is the number of
+its vertices minus one. A path is induced if there are no graph edges joining non-consecutive vertices.
+In a directed graph, a directed path is a path in which all arcs are oriented in the same direction. For a
+path P of a graph G between two vertices u and v, the vertices V (P) \ {u, v} are internal vertices of P.
+A path between two vertices u and v is called a (u, v)-path. Similarly, we have the notions of isometric
+(u, v)-path and induced (u, v)-path. For a vertex r of G and a set S of vertices of G, the distance of S from
+r, denoted as d (r, S), is the minimum of the distance between any vertex of S and r. For a subgraph
+H of G, the distance of H w.r.t. r is d (r, V (H)). Formally, we have d (r, S) = min{d (r, v) : v ∈ S} and
+d (r, H) = d (r, V (H)).
+4
+
+For a graph G and a vertex r ∈ V (G), consider the following operations on G. First, remove all
+edges xy from G such that d (r, x) = d (r, y).
+Let G′
+r be the resulting graph.
+Then, for each edge
+e = xy ∈ E(G′
+r) with d (r, x) = d (r, y) − 1, orient e from y to x. Let −→
+Gr be the directed acyclic graph
+formed after applying the above operation on G′. Note that this digraph can easily be computed in linear
+time using a Breadth-First Search (BFS) traversal with starting vertex r.
+The following definition is inspired by the terminology of posets (as the graph −→
+Gr can be seen as the
+Hasse diagram of a poset).
+Definition 7. For a graph G and a vertex r ∈ V (G), two vertices x, y ∈ V (G) are antichain vertices if
+there are no directed paths from x to y or from y to x in −→
+Gr. A set X of vertices of G is an antichain
+set if any two vertices in X are antichain vertices. The cardinality of the largest antichain set in −→
+Gr will
+be denoted by β
+�−→
+Gr
+�
+. The cardinality of the largest antichain set of G, is defined as
+β (G) = min
+�
+β
+�−→
+Gr
+�
+: r ∈ V (G)
+�
+Definition 8 ([7]). Let r be a vertex of a graph G. For a subgraph H, Ar (H) shall denote the maximum
+antichain set of H in −→
+Gr. The isometric path antichain cover number of −→
+Gr, denoted by ipacc
+�−→
+Gr
+�
+, is
+defined as follows:
+ipacc
+�−→
+Gr
+�
+= max {|Ar (P) |: P is an isometric path}
+The isometric path antichain cover number of graph G, denoted as ipacc (G), is defined as the minimum
+over all possible antichain covers of its associated directed acyclic graphs:
+ipacc (G) = min
+�
+ipacc
+�−→
+Gr
+�
+: r ∈ V (G)
+�
+We recall the proof of the following proposition from [7] which will be used heavily in this paper.
+Proposition 9 ([7]). Let G be a graph and r, an arbitrary vertex of G. Consider the directed acyclic
+graph −→
+Gr, and let P be an isometric path between two vertices x and y in G. Then |P| ≥ |d (r, x) −
+d (r, y) | + |Ar (P) | − 1.
+Proof. Orient the edges of P from y to x in G. First, observe that P must contain a set E1 of oriented
+edges such that |E1| = |d (r, y) − d (r, x) | and for any −→
+ab ∈ E1, d (r, a) = d (r, b) + 1. Let the vertices of
+the largest antichain set of P in −→
+Gr, i.e., Ar (P), be ordered as a1, a2, . . . , at according to their occurrence
+while traversing P from y to x. For i ∈ [2, t], let Pi be the subpath of P between ai−1 and ai. Observe that
+for any i ∈ [2, t], since ai and ai−1 are antichain vertices, there must exist an oriented edge −→
+bici ∈ E(Pi)
+such that either d (r, bi) = d (r, ci) or d (r, bi) = d (r, ci) − 1.
+Let E2 = {bici}i∈[2,t].
+Observe that
+E1 ∩ E2 = ∅ and therefore |P| ≥ |E1| + |E2| = |d (r, y) − d (r, x) | + |Ar (P) | − 1.
+3
+Proof of Theorem 2
+In this section, we shall show that isometric path antichain cover number of graphs with hyperbolicity
+at most δ is at most 12δ + 6. To achieve our goal we need to recall a few definitions from the literature.
+For three vertices x, y, z of a graph G, a geodesic triangle [3], denoted as ∆(x, y, z) is the union P(x, y) ∪
+P(y, z)∪P(x, z) of three isometric paths connecting these vertices. A geodesic triangle ∆(x, y, z) is called
+ρ-slim if for any vertex u ∈ P(x, y) the distance d (u, P(y, z) ∪ P(x, z)) is at most ρ. The smallest value
+of ρ for which every geodesic triangle of G is ρ-slim is called the slimness of G and is denoted by sl (G).
+In the following lemma, we shall show that if the isometric path antichain cover number of a graph is
+large then so is the slimness of the graph.
+Lemma 10. For any graph G, ipacc (G) ≤ 4sl (G) + 2.
+5
+
+u
+v
+c
+c′
+Figure 3: An example of a 4-fat turtle. Let C be the cycle induced by the black vertices, P be the path
+induced by the white vertices. Then the tuple (4, C, P, c, c′) defines a 4-fat turtle.
+Proof. Let ρ = sl (G).
+Aiming for a contradiction, let r be a vertex of G such that there exists an
+isometric path P such that |Ar (P) | ≥ 4ρ + 3. Let the vertices of Ar (P) be named and ordered as
+a1, a2, . . . , a2ρ+2, . . . , a4ρ+3 as they are encountered while traversing P from one end-vertex to the other.
+Let x = a1, y = a4ρ+3. Let −→
+Px be an oriented path from x to r in −→
+Gr. Observe that Px, the path of
+G obtained by removing the orientation of −→
+Px, is an (x, r)-isometric path. Let −→
+Py be an oriented path
+from y to r in −→
+Gr. Similarly, Py, the path of G obtained by removing the orientation of −→
+Py, is an (y, r)-
+isometric path. Observe that P, Px, Py form a geodesic triangle with x, r, y as end-vertices. Consider the
+vertex z = a2ρ+2 on the path P. Since ρ = sl (G), there exists a vertex w ∈ V (Px) ∪ V (Py) such that
+d (w, z) ≤ ρ. Without loss of generality, assume w ∈ V (Px). Then, d (x, z) ≤ d (x, w) + d (w, z). By
+using that d (r, z) ≤ d (r, w) + d (w, z) ≤ d (r, w) + ρ, we get d (x, z) ≤ |d (r, x) − d (r, z) | + 2ρ. But this
+contradicts Proposition 9, due to which we have d (x, z) ≥ |d (r, x) − d (r, z) | + 2ρ + 1.
+Now we shall use the following result.
+Proposition 11 ([3]). For any graph G, sl (G) ≤ 3hb (G).
+Proposition 11 and Lemma 10, imply the theorem.
+4
+Proofs of Theorem 3 and 5
+In this section, we shall prove Theorems 3 and 5. First we shall define the notions of t-theta, t-prism,
+and t-pyramid [29].
+For an integer t ≥ 1, a t-prism is a graph made of three vertex-disjoint induced paths P1 = a1 . . . b1,
+P2 = a2 . . . b2, P3 = a3 . . . b3 of lengths at least t, such that a1a2a3 and b1b2b3 are triangles and no edges
+exist between the paths except those of the two triangles. For an integer t ≥ 1, a t-pyramid is a graph
+made of three induced paths P1 = a . . . b1, P2 = a . . . b2, P3 = a . . . b3 of lengths at least t, two of which
+have lengths at least t + 1, they are pairwise vertex-disjoint except at a, such that b1b2b3 is a triangle
+and no edges exist between the paths except those of the triangle and the three edges incident to a. For
+an integer t ≥ 1, a t-theta is a graph made of three internally vertex-disjoint induced paths P1 = a . . . b,
+P2 = a . . . b, P3 = a . . . b of lengths at least t+1, and such that no edges exist between the paths except the
+three edges incident to a and the three edges incident to b. A graph G is (t-theta, t-pyramid, t-prism)-free
+if G does not contain any induced subgraph isomorphic to a t-theta, t-pyramid or t-prism. When t = 1,
+(t-theta, t-pyramid, t-prism)-free graphs are exactly (theta, prism, pyramid)-free graphs.
+Now we shall show that the isometric path antichain cover number of (t-theta, t-pyramid, t-prism)-
+free graphs are bounded above by a linear function on t. We shall show that, when the isometric path
+antichain cover number of a graph is large, the existence of a structure called “t-fat turtle” (defined
+below) as an induced subgraph is forced, which, cannot be present in a ((t − 1)-theta, (t − 1)-pyramid,
+(t − 1)-prism)-free graph.
+Definition 12. For an integer t ≥ 1, a “t-fat turtle” consists of a cycle C and an induced (u, v)-path P
+of length at least t such that all of the following hold.
+6
+
+(a) V (P) ∩ V (C) = ∅,
+(b) For any vertex w ∈ (V (P) \ {u, v}), N(w) ∩ V (C) = ∅ and both u and v have at least one neighbour
+in C.
+(c) For any vertex w ∈ N(u) ∩ V (C) and w′ ∈ N(v) ∩ V (C), the distance between w and w′ in C is at
+least t,
+(d) There exist two vertices {c, c′} ⊂ V (C) and two distinct components Cu, Cv of C − {c, c′} such that
+N(u) ∩ V (C) ⊆ V (Cu) and N(v) ∩ V (C) ⊆ V (Cv).
+The tuple (t, C, P, c, c′) defines the t-fat turtle. See Figure 3 for an example.
+In the following observation, we show that any (t-theta, t-pyramid,t-prism)-free graph cannot contain
+a (t + 1)-fat turtle as an induced subgraph.
+Lemma 13. For some integer t ≥ 1, let G be a graph containing a (t + 1)-fat turtle as an induced
+subgraph. Then G is not (t-theta, t-pyramid, t-prism)-free.
+Proof. Let (t+1, C, P, c, c′) be a (t+1)-fat turtle in G. Let the vertices of C be named c = a0, a1, . . . , ak =
+c′, ak+1, . . . , a|V (C)| as they are encountered while traversing C starting from c in a counter-clockwise
+manner. Denote by u, v the end-vertices of P. By definition, there exist two distinct components Cu, Cv of
+C−{c, c′} such that N(u)∩V (C) ⊆ V (Cu) and N(v)∩V (C) ⊆ V (Cv). Without loss of generality, assume
+V (Cu) = {a1, a2, . . . , ak−1} and V (Cv) = {ak+1, ak+2, . . . , a|V (C)|}. Let i− and i+ be the minimum and
+maximum indices such that ai− and ai+ are adjacent to u. Let j− and j+ be the minimum and maximum
+indices such that aj− and aj+ are adjacent to v. By definition, i− ≤ i+ < j− ≤ j+. Let P1 be the
+(ai−, aj+)-subpath of C containing c. Let P2 be the (ai+, aj−)-subpath of C that contains c′. Observe
+that P1 and P2 have length at least t (by definition). Now we show that P, P1, P2 together form one of
+theta, pyramid or prism. If ai− = ai+ and aj− = aj+, then P, P1, P2 form a t-theta. If i− ≤ i+ − 2 and
+j− ≤ j+ − 2, then also P, P1, P2 form a t-theta. If j− = j+ − 1 and i− = i+ − 1, then P, P1, P2 form a
+t-prism. In any other case, P, P1, P2 form a t-pyramid.
+In the remainder of this section, we shall prove that there exists a linear function f(t) such that if
+the isometric path antichain cover number of a graph is more than f(t), then G is forced to contain a
+(t + 1)-fat turtle as an induced subgraph, and therefore is not (t-theta, t-pyramid,t-prism)-free. We shall
+use the following observation.
+Observation 14. Let G be a graph, r be an arbitrary vertex, P be an isometric (u, v)-path in G and Q
+be a subpath of an isometric (v, r)-path in G such that one endpoint of Q is v. Let P ′ be the maximum
+(u, w)-subpath of P such that no internal vertex of P ′ is a neighbour of some vertex of Q. We have that
+|Ar (P ′) | ≥ |Ar (P) | − 3.
+Proof. Suppose |Ar (P ′) | ≤ |Ar (P) | − 4 and consider the (w, v)-subpath, say P ′′, of P. Observe that
+|Ar (P ′′) | ≥ 4. Now let w′ be a vertex of Q which is a neighbour of w. Observe that |d (r, w)−d (r, w′) | ≤ 1
+and therefore d (w, v) = |E(P ′′)| ≤ |d (r, w)−d (r, v) |+2. But this contradicts Proposition 9, which implies
+that the length of P ′′ is at least |d (r, w) − d (r, v) | + 3.
+Lemma 15. For an integer t ≥ 1, let G be a graph with ipacc (G) ≥ 8t + 64. Then G has a (t + 1)-fat
+turtle as an induced subgraph.
+Proof. Let r be a vertex of G such that ipacc
+�−→
+Gr
+�
+is at least 8t+64. Then there exists an isometric path
+P such that |Ar (P) | ≥ 8t+ 64. Let the two endpoints of P be a and b. (See Figure 4.) Let u be a vertex
+of P such that d (r, u) = d (r, P). Let Pau be the (a, u)-subpath of P and Pbu be the (b, u)-subpath of P.
+Both Pau and Pbu are isometric paths and observe that either |Ar (Pau) | ≥ 4t+32 or |Ar (Pbu) | ≥ 4t+32.
+Without loss of generality, assume that |Ar (Pbu) | ≥ 4t + 32. Let Qr
+b be an isometric (b, r)-path in G.
+7
+
+r
+z2
+w2
+u
+z
+z1
+w
+b
+w1
+c
+(= a2t+13)
+x
+c1
+a
+c2
+T (c1, c2)
+≥ t
+≥ t
+≥ t
+Qr
+b
+Qr
+u
+Figure 4: Illustration of the notations used in the proof of Lemma 15.
+Let Ruw be the maximum (u, w)-subpath, of Pbu such that no internal vertex of Ruw is a neighbour
+of Qr
+b. Note that Ruw is an isometric path and w has a neighbour in Qr
+b. Applying Observation 14, we
+have the following:
+Claim 15.1. |Ar (Ruw) | ≥ 4t + 29.
+Let Qr
+u be any isometric (u, r)-path of G and let Rzw be the maximum (z, w)-subpath of Ruw such
+that no internal vertex of Rzw has a neighbour in Qr
+u. Observe that Rzw is an isometric path, and z has
+a neighbour in Qr
+u. Again applying Observation 14, we have the following:
+Claim 15.2. |Ar (Rzw) | ≥ 4t + 26.
+Let a1, a2, . . . , ak be the vertices of Ar (Rzw) ordered according to their appearance while traversing
+Rzw from z to w. Due to Claim 15.2, we have that k ≥ 4t + 26. Let c = a2t+13 and Qr
+c denote an
+isometric (c, r)-path. Let T (r, c1) be the maximum subpath of Qr
+c such that no internal vertex of T (r, c1)
+is adjacent to any vertex of Rzw.
+Claim 15.3. Let x be a neighbor of c1 in Rzw, X be the (x, b)-subpath of Pub and Y be the (x, u)-subpath
+of Pub. Then |Ar (X) | ≥ 2t + 11 and |Ar (Y ) | ≥ 2t + 11.
+Proof. Let Rcw denote the (c, w)-subpath of Rzw. Observe that |Ar (Rcw) | ≥ 2t + 14. First, consider
+the case when x lies in the (z, c)-subpath of Rzw. In this case, Rcw is a subpath of X and therefore
+|Ar (X) | ≥ 2t + 14. Now consider the case when x lies in Rcw. In this case, applying Observation 14,
+we have that |Ar (X) | ≥ |Ar (Rcw) | − 3 ≥ 2t + 11. Using a similar argument, we have that |Ar (Y ) | ≥
+2t + 11.
+Let T (c1, c2) be the maximum (c1, c2)-subpath of T (c1, r) such that no internal vertex of T (c1, c2) is
+adjacent to a vertex of Qr
+b or Qr
+u. We have the following claim.
+Claim 15.4. The length of T (c1, c2) is at least t + 3.
+Proof. Assume that the length of T (c1, c2) is at most t + 2 and x be a neighbour of c1 in Rzw. Observe
+that all vertices of Rzw are at distance at least d (r, u) i.e. d (r, Rzw) ≥ d (r, u), since d (r, u) = d (r, P).
+Hence,
+(+) d (r, x) ≥ d (r, u) and d (r, c1) ≥ d (r, u) − 1.
+8
+
+Now, suppose c2 has a neighbor c3 in Qr
+u. Hence d (c3, x) ≤ d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4.
+Now, using (+) and the fact that c3 lies on an isometric (r, u)-path (Qr
+u), we have that d (c3, u) ≤ t + 4.
+Therefore, d (u, x) ≤ d (c3, u) + d (c3, x) ≤ 2t + 8. But this contradicts Proposition 9 and Claim 15.3, as
+they together imply that d (u, x) is at least d (r, x) − d (r, u) + 2t + 10≥ 2t + 10.
+Hence, c2 must have a neighbour c3 in Qr
+b. First, assume that d (r, x) ≥ d (r, b). Then, as d (c3, x) ≤
+d (c3, c2) + d (c2, c1) + d (c1, x) ≤ t + 4 and c3 lies on an isometric (r, b)-path (Qr
+b), we have that d (x, b) ≤
+2t + 8. But again this contradicts Proposition 9 and Claim 15.3, as they together imply that the length
+of d (x, b) is at least d (r, x) − d (r, u) + 2t + 10. Now, assume that d (r, x) < d (r, b). Let b′ be a vertex of
+Qr
+b such that d (r, b′) = d (r, x). Using a similar argumentation as before, we have that d (x, b′) ≤ 2t + 8.
+Hence, d (x, b) ≤ d (x, b′) + d (b′, b) ≤ d (r, b) − d (r, x) + 2t + 8. But this contradicts Proposition 9 which,
+due to Claim 15.3, implies that d (x, b) ≥ d (r, b) − d (r, x) + 2t + 10.
+The path T (c1, c2) forms the first ingredient to extract a (t + 1)-fat turtle. Let z1 be the neighbor of
+z in Qr
+u and w1 be the neighbour of w in Qr
+b. We have the following claim.
+Claim 15.5. The vertices w1 and z1 are non adjacent.
+Proof. Recall that z1 lies in Qr
+u and d (r, z) ≥ d (r, u). Hence z1 must be a neighbor of u. If w1 and z1 are
+adjacent, then observe that d (u, b) ≤ d (r, b) − d (r, w1) + 2 ≤. This implies d (u, b) ≤ d (r, b) − d (r, u)+ 3.
+But this shall again contradict Proposition 9.
+Now we shall construct a (w1, z1)-path as follows: Consider the maximum (w1, w2)-subpath, say
+T (w1, w2), of Qr
+b such that no internal vertex of T (w1, w2) has a neighbour in Qr
+u. Similarly, consider the
+maximum (z1, z2)-subpath, say T (z1, z2), of Qr
+b such that no internal vertex of T (z1, z2) is a neighbor of
+w2. Let T be the path obtained by taking the union of T (w1, w2) and T (z1, z2). Observe that z2 must
+be a neighbour of w2 and T is an induced (w1, z1)-path. The definitions of T and Rzw imply that their
+union induces a cycle Z. Here we have the second and final ingredient to extract the (t + 1)-fat turtle.
+Suppose that c2 has a neighbour in T . Let T ′ be the maximum subpath of T (c1, c2) which is vertex-
+disjoint from Z. Due to Claim 15.4, the length of T ′ is at least t + 1. Let e1 and e2 be the end-vertices
+of T ′. Observe the following.
+• Each of e1 and e2 has at least one neighbor in Z.
+• Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).
+• For a vertex e′
+1 ∈ N(e1) ∩ V (Z) and e′
+2 ∈ N(e2) ∩ V (Z), the distance between e′
+1 and e′
+2 is at least
+t + 1. This statement follows from Claim 15.3.
+Hence, we have that the tuple (t + 1, Z, T ′, z, w) defines a (t + 1)-fat turtle. Now consider the case
+when c2 does not have a neighbor in T . By definition, c2 has at least one neighbor in Qr
+u or Qr
+b. Without
+loss of generality, assume that c2 has a neighbor c3 in Qr
+u such that the (z2, c3)-subpath, say, T ′′ of Qr
+u
+has no neighbor of c2 other than c3. Observe that the path T ∗ = (T ′ ∪ (T ′′ − {z2})) is vertex-disjoint
+from Z and has length at least t + 1. Let e1, e2 be the two end-vertices of T ∗. Observe the following.
+• Each of e1 and e2 has at least one neighbor in Z.
+• Z −{z, w} contains two distinct components C1, C2 such that for i ∈ {1, 2}, N(ei)∩V (Z) ⊆ V (Ci).
+• For a vertex e′
+1 ∈ N(e1) ∩ V (Z) and e′
+2 ∈ N(e2) ∩ V (Z), the distance between e′
+1 and e′
+2 is at least
+t + 1. This statement follows from Claim 15.3.
+Hence, (t + 1, Z, T ∗, z, w) is a (t + 1)-fat turtle
+Proof of Theorem 5 and 3: Lemma 13 and 15 together imply the theorems.
+9
+
+5
+Proof of Theorem 4
+Next we shall show that outerstring graphs are (4-theta, 4-prism, 4-pyramid)-free.
+Lemma 16. Let G be an outerstring graph. Then, G is (4-theta, 4-prism, 4-pyramid)-free.
+Proof. To prove the lemma, we shall need to recall a few definitions and results from the literature. A
+graph G is a string graph if there is a collection S of simple curves on the plane and a bijection between
+V (G) and S such that two curves in S intersect if and only if the corresponding vertices are adjacent in
+G. Let G be a graph with an edge e. The graph G \ e is obtained by contracting the edge e into a single
+vertex. Observe that string graphs are closed under edge contraction [22]. We shall use the following
+result.
+Proposition 17 ([22]). Let G be an outerstring graph with an edge e. Then G\e is an outerstring graph.
+A full subdivision of a graph means replacing each edge of G with a new path of length at least two.
+We shall use the following result implied from Theorem 1 of [22].
+Proposition 18 ([22]). Let G be a string graph. Then G does not contain a full subdivision of K3,3 as
+an induced subgraph.
+For a graph G, the graph G+ is constructed by introducing a new apex vertex a and connecting a
+with all vertices of G by new copies of paths of length at least 2. We shall use the following result of
+Biedl et al. [4].
+Proposition 19 (Lemma 1, [4]). A graph G is an outerstring graph if and only if G+ is a string graph.
+Now we are ready to prove the lemma.
+Let G be an outerstring graph. Assume for the sake of
+contradiction that G contains an induced subgraph H which is a 4-theta, 4-pyramid, or a 4-prism. Since
+every induced subgraph of an outerstring graph is also an outerstring graph, we have that H is an
+outerstring graph. Let E be the set of edges of H whose both endpoints are part of some triangle. Now
+consider the graph H1 = H \ E which is obtained by contracting all edges in E. By Proposition 17, H1
+is an outerstring graph and it is easy to check that H1 is a 3-theta. Let u and v be the vertices of H1
+with degree 3 and w1, w2, w3 be the set of mutually non-adjacent vertices such that for each i ∈ {1, 2, 3}
+d (u, wi) = 2 and d (v, wi) ≥ 2. Since H1 is a 3-theta, w1, w2, w3 exist. Now consider the graph H+
+1 and
+a be the new apex vertex. Due to Proposition 19, we have that H+
+1 is a string graph. But notice that,
+for each pair of vertices in {x, y} ⊂ {w1, w2, w3, u, v, a}, there exists a unique path of length at least 2
+connecting x, y. This implies that H+
+1 (which is a string graph) contains a full subdivision of K3,3, which
+contradicts Proposition 18.
+Proof of Theorem 4: Lemma 16 and Theorem 5 together imply the theorem.
+6
+Conclusion
+In this paper, we derived upper bounds on the isometric path antichain cover number of three seemingly
+(structurally) different classes of graphs, namely hyperbolic graphs, (theta,pyramid,prism)-free graphs
+and outerstring graphs. We have not made any efforts in reducing the constants in our bounds. In
+particular, we believe that a careful analysis of the structure of outerstring graphs would help in reducing
+its isometric path antichain cover number. (Note that outerstring graphs may contain a theta, 2-pyramid
+or a 2-prism). We note that the isometric path antichain cover number of a (n × n)-grid is Ω(n), which
+implies that the isometric path antichain cover number of planar graphs (which are also string graphs)
+is not bounded. Similarly, we note that the isometric path antichain cover number of G1, G2 and G3
+are unbounded where G1 denotes the class of (theta, prism)-free graphs, G2 denotes the class of (prism,
+pyramid)-free graphs and G3 denotes the class of (theta, pyramid)-free graphs. An interesting direction
+10
+
+of research is to generalise the properties of hyperbolic graphs to graphs with bounded isometric path
+antichain cover number.
+We also note that recognizing graphs with a given value of isometric path antichain cover number
+might be computationally hard. This problem does not seem to be in NP: to certify that a graph has
+isometric path antichain cover number at most k, (intuitively) one would need to check, for all possible
+isometric paths, that it does not contain any antichain of size k + 1 (with respect to all possible roots r).
+On the contrary, it is in coNP: to certify that its isometric path antichain cover number is not at most k,
+one may exhibit, for every possible root r, one isometric path and one antichain of size k + 1 contained
+in the path. Checking the validity of this certificate can be done in polynomial time. We do not know if
+the problem is coNP-hard. Nevertheless, this parameter seems interesting from a structural graph theory
+point of view, since it encapsulates several seemingly unrelated graph classes with, as a consequence,
+common algorithmic behaviours of these classes (recall that the value of the parameter does not need to
+be computed for the approximation algorithm to work). Using our framework, perhaps other common
+properties of these classes could be exhibited?
+Our results imply a constant factor approximation algorithm for Isometric Path Cover on hyper-
+bolic graphs, (theta, pyramid, prism)-free graphs and outerstring graphs. However, the existence of a
+constant factor approximation algorithm for Isometric Path Cover on general graphs is not known
+(it was observed that the algorithm from [7] also used here, can have non-constant approximation ratios,
+for example on hypercube graphs, whose isometric path antichain cover numbers are unbounded).
+Polynomial-time solvability of Isometric Path Cover on restricted graph classes like split graphs,
+interval graphs, planar graphs etc. also remains unknown, see [7].
+Acknowledgement: We thank Nicolas Trotignon for suggesting us to study the class of (t-theta, t-
+pyramid, t-prism)-free graphs.
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+

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+STOCHASTIC APPROACHES: MODELING THE PROBABILITY OF
+ENCOUNTERS BETWEEN H2-MOLECULES AND METALLIC
+ATOMIC CLUSTERS IN A CUBIC BOX
+Maximiliano L. Riddick, Leandro Andrini
+Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas
+Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET)
+Departamento de Matemática, Fac. de Ciencias Exactas, UNLP
+La Plata, Argentina
+mriddick@mate.unlp.edu.ar
+Enrique E. Álvarez
+Instituto de Cálculo, Fac. Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón II, UBA (CABA)
+Departamento de Fisicomatemática, Fac. de Ingeniería, UNLP
+Ciudad Autónoma de Buenos Aires and La Plata, Argentina
+Félix G. Requejo
+Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas
+Departamento de Química, Fac. de Ciencias Exactas (INIFTA/ UNLP-CONICET)
+Departamento de Física, Fac. de Ciencias Exactas, UNLP
+La Plata, Argentina
+ABSTRACT
+In recent years the advance of chemical synthesis has made it possible to obtain “naked”clusters of
+different transition metals. It is well known that cluster experiments allow studying the fundamental
+reactive behavior of catalytic materials in an environment that avoids the complications present in
+extended solid-phase research. In physicochemical terms, the question that arises is the chemical
+reduction of metallic clusters could be affected by the presence of H2 molecules, that is, by the
+probability of encounter that these small metal atomic agglomerates can have with these reducing
+species. Therefore, we consider the stochastic movement of N molecules of hydrogen in a cubic
+box containing M metallic atomic clusters in a confined region of the box. We use a Wiener process
+to simulate the stochastic process, with σ given by the Maxwell-Boltzmann relationships, which
+enabled us to obtain an analytical expression for the probability density function. This expression is
+an exact expression, obtained under an original proposal outlined in this work, i.e. obtained from
+considerations of mathematical rebounds. On this basis, we obtained the probability of encounter
+for three different volumes, 0.1
+3, 0.2
+3 and 0.4
+3 m
+3, at three different temperatures in each case,
+293, 373 and 473 K, for 10
+1 ≤ N ≤ 10
+10, comparing the results with those obtained considering
+the distribution of the position as a Truncated Normal Distribution. Finally, we observe that the
+probability is significantly affected by the number N of molecules and by the size of the box, not by
+the temperature.
+Keywords Wiener Process · Probability of encounters · Molecular Collisions · Atomic-Clusters · Mathematical
+Rebounds
+arXiv:2301.13797v1  [cond-mat.mtrl-sci]  10 Jan 2023
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+1
+Introduction
+In the last two decades there has been an important development in clusters chemistry, and consequently new questions
+arise on the basis of these developments [1, 2, 3, 4, 5, 6, 7]. This interest is due to an atomic clusters containing
+up to a few dozen atoms exhibit features that are very different from the corresponding bulk properties and that can
+depend very sensitively on cluster size [8]. In particular, many of these transition metal clusters are used in the field of
+catalysis [1, 9, 10, 11]. One of the basic principles of catalysis is that when the smaller the metal particles, the larger the
+fraction of the metal atoms that are exposed at surfaces, where they are accessible to reactant molecules and available
+for catalysis [1]. It is well known in chemistry that the encounter between two molecules can give rise to a chemical
+reaction, and from the mathematical aspect there are two fundamental ways to represent these types of situations as
+continuous, represented by differential equations whose variables are concentrations, or as discrete, represented by
+stochastic processes whose variables are the number of molecules [12].
+Without loss of generality, it can be considered that the molecular chemisorption is due to the encounter between
+a molecule and a surface (or a cluster in this case) with the energy necessary for the phenomenon of adsorption to
+occur [13]. Besides, the kinetics of hydrogen chemisorption by neutral gas-phase metal clusters exhibits a complex
+dependence on both cluster size and metal type [14]. For different chemical purposes, for example, in the case of copper
+clusters (Cun) is very important to have control of the chemisorption of hydrogen on these clusters, i.e. the formation of
+Cun-H2 species [15].
+From a reductionist point of view, the molecular chemisorption is a problem of encounter between bodies: metal clusters
+and reactant molecules. In our first approximation (mathematical reduction) we will consider the problem as a problem
+of encounter or collisions between bodies. We are interested in proposing this strategy because we are focused to answer
+what is the probability of meeting between N hydrogen molecules (N-H2) and a fixed M metallic clusters (M-Men),
+for a given time t, where the H2 move freely in a bounded volume V of R3-space. Under this assumption, we are going
+to consider H2-molecules and Men-clusters as rigid spheres of radii r1 and r2, respectively. Then, it is considered that
+there will be a collision whenever the center-to-center distance between an H2-molecule and a Men-cluster is equal to
+r12 = r1 + r2 [16]. Also, in this context we propose the H2-molecules follow a Brownian motion, namely: (a) it has
+continuous trajectories (sample paths) and (b) the increments of the paths in disjoint time intervals are independent zero
+mean Gaussian random variables with variance proportional to the duration of the time interval [17].
+The pioneering work of T.D. Gillespie [16, 18, 19] have given rise to a large number of works that are proposed different
+algorithms for the calculation for numerically simulating the time evolution of a well-stirred chemically reacting system,
+although despite recent major improvements in the efficiency of the stochastic simulation algorithm, its drawback
+remains the great amount of computer time that is often required to simulate a desired amount of system time [20]. While
+our method is a simple reduction to collisions of molecules, allows to calculate the probability of encounter (scheduled
+in R) for a large number of molecules (≈ 106) and clusters (≈ 1020) with advantages regarding the cost of calculation,
+and the effects of first approximation can provide statistical support to the design of experiments. This calculation
+is possible using a stochastic model (Wiener process) in the context of considerations from the Maxwell-Boltzmann
+theory.
+2
+A first theoretical approaching
+As we announced in the introduction, we will assume that hydrogen molecules have a random movement, whence
+let H(t) = (X(t), Y (t), Z(t)) the random variable which specify the space point where the H2 hydrogen molecule
+is at time t. Trivially, H(t) depends on an initially point H(0) = (x0, y0, z0). Thus, when the initial starting point is
+undefined, H(t) = H(t, x0, y0, z0). Our interest is in how probably is that the distance between H(t) and a fixed point
+(a, b, c) is smaller than ϵ. The fixed point (a, b, c) are the coordinates for Men.
+Let’s consider the random variable D(t) as the variable that measures the distance between H(t) and the fixed point
+(a, b, c). Following the classical Pythagorean relationship, D(t) =
+�
+(X(t) − a)2 + (Y (t) − b)2 + (Z(t) − c)2, and
+in general D(t) = D(t, x0, y0, z0, a, b, c).
+Now, given a time window [0, τ], let
+Rτ :=
+�
+�
+�
+1
+if min
+t∈[0,τ) D(t) ≤ ϵ,
+0
+otherwise.
+(1)
+So, for a fixed t0 > 0, we define G(t0) = P(D(t0) ≤ ϵ) =
+� ϵ
+0 D(s)ds. Then, P(Rτ = 1) =
+� τ
+0 G(t)dt.
+2
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Thus, given τ > 0, Rτ depends only on the initial values (x0, y0, z0, a, b, c). Now, if we have M-Men, the probability
+that the H2 molecule does not meet with any of the clusters is P(Rτ1 = 0, Rτ2 = 0, ..., RτM = 0) = pA, where
+Rτi, i ∈ {1, ..., M}, follows the definition given in the eq. 1.
+If N-H2 molecules are in the environment, let Aj the event “the j-th hydrogen molecule meet with a metallic cluster”.
+Under random starting points, we are interested in P(AC
+1 ∩ AC
+2 ∩ ... ∩ AC
+N) = pN
+A according to the independence
+among the hydrogen molecules.
+2.1
+Adaptation to our context
+Next, we proceed to realize the analysis according to the Brownian Motion Theory [17], in which the movement of
+the particle is independent among different axis, and we are going to assume that it follows a Wiener process [21, 22].
+Then,
+X(t) = x0 + WX(t)
+Y (t) = y0 + WY (t)
+Z(t) = z0 + WZ(t)
+And we will say that WX(t), WY (t) and WZ(t) are following a Wiener processes with σ =
+�
+kbT
+m , where kb is the
+Boltzmann’s constant, T is the absolute temperature in Kelvin (K) and m is the H2’s mass in kg. That is, we are
+imposing a physical behavior that obeys Maxwell-Boltzmann’s considerations. According this:
+X(t) ∼ N(x0, σ2t)
+Y (t) ∼ N(y0, σ2t)
+Z(t) ∼ N(z0, σ2t)
+With density function fX(x, t|x0), fY (y, t|y0) and fZ(z, t|z0), respectively. Under these assumptions:
+fX(x, t|x0) =
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�x − x0
+σ
+√
+t
+�2�
+fY (y, t|y0) =
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�y − y0
+σ
+√
+t
+�2�
+fZ(z, t|z0) =
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�z − z0
+σ
+√
+t
+�2�
+2.1.1
+Unbounded conditions
+Under unbounded conditions, as it is well known, the density of the particle position in the space for a fixed t follows
+the expression:
+fXY Z(x, y, z, t|x0, y0, z0) = fX(x, t|x0) · fY (y, t|y0) · fZ(z, t|z0) =
+=
+1
+(
+�
+2πσ2t)3 exp
+�
+−1
+2
+�(x − x0)2 + (y − y0)2 + (z − z0)2
+σ2t
+��
+This function is continuous in the variables x, y, z, t, then is also integrable in a measurable context. Because of this
+fact, Fubini’s theorem is aplicable. Now, calling ν =
+�
+(x−x0)2+(y−y0)2+(z−z0)2
+2σ2
+, and integrating over the variable t by
+substitution, results:
+3
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+fXY Z(x, y, z, τ|x0, y0, z0) =
+1
+−ν(
+√
+2πσ2)3
+� τ
+0
+−ν
+(
+√
+t)3 exp
+�
+−
+� ν
+√
+t
+�2�
+dt
+=
+1
+ν(
+√
+2πσ2)3
+� ∞
+√ ν
+τ
+e−u2du
+Remembering that the erfc function [23] is defined by:
+erfc(z) =
+2
+√π
+� ∞
+z
+e−t2dt
+we conclude:
+fXY Z(x, y, z, τ|x0, y0, z0) =
+1
+2πν(
+√
+2σ2)3 erfc
+��ν
+τ
+�
+From the physical-experimental perspective that the problem is lays out, the unbounded system lacks interest, so we
+will proceed to study the case of the bounded system.
+2.1.2
+Bounded conditions
+We assume that the experiment takes place into a cubic recipe centered at the origin. This implies that X(t), Y (t) and
+Z(t) ∈ [−L; L], for a fixed volume V = L3 in R3-space.
+In a similar issue the traditional way of approaching is by “truncation" [24, 25]. A drawback of this approach is the
+fact that the truncation does not represent precisely the reflection on the boundaries. An illustrative and motivational
+argument is given by the following example: suppose a random walk of N = 4 steps, with starting point at the origin.
+Then, the walker moves 1 step at right or left (with equal probability) at each step. Then, after four steps, the resultant
+probabilities of the walker position are:
+0; with probability 3/8,
+−2 or 2; with probability 2/8,
+−4 or 4; with probability 1/16.
+The probability values (under truncation) in the closed interval [−2, 2] for the values (−2, −1, 0, 1, 2) are, respectively:
+(2/7, 0, 3/7, 0, 2/7)
+With fixed boundaries, considering reflections at [−2, 2], we can construct the following Markov transition matrix P:
+P =
+0
+1
+0
+0
+0
+1/2
+0
+1/2
+0
+0
+0
+1/2
+0
+1/2
+0
+0
+0
+1/2
+0
+1/2
+0
+0
+0
+1
+0
+At the fourth step, after some algebra, we obtain the respectively mass point probability for the position of the walker.
+This is provided by the stochastic vector
+(1/4, 0, 1/2, 0, 1/4)
+(given by the third file of P 4, i.e.: with starting point at the origin). At this point, is clearly the difference between
+truncation and “rebounds" (considering reflection on the boundary).
+We must modify the density of the position H(t) according to the particle rebounds (see Fig. 1). It is important to note
+that the rebounds indicated in the figure in gray colour do not correspond to the physical rebounds of the particles in the
+cubic box, but to the contributions of the displaced distribution considering an infinite behavior.
+4
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Figure 1: In red colour an arbitrary normal distribution, N(0, σ). We observe in gray colour the folding of the normal
+distribution at the edge of the box. We could see that A + B = 2L.
+Inside the box, the derived density fB according to the variable X(t) ∼ N(x0, σ2t) with density function fX of the
+particle position (for each dimension, see Fig. 1) follows the expression:
+fB(x) = [fX(x) + fB+(x) + fB−(x)] × I[−L;L](x)
+where:
+fB+(x) = f(x + 2A) + f(x + 2A + 2B) + f(x + 2A + 2B + 2A) + ... =
+= f(x + 2(L − x)) + f(x + 2(L − x) + 2(x − (−L)) + ...
+= f(−x + 2L) + f(x + 4L) + f(−x + 6L) + ...
+=
+∞
+�
+k=1
+f((−1)kx + 2kL)
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx + 2kL) − x0
+σ
+√
+t
+�2�
+and
+fB−(x) = f(x − 2B) + f(x − 2B − 2A) + f(x − 2B − 2A − 2B) + ... =
+= f(x − 2(x − (−L))) + f(x − 2(x − (−L)) − 2(L − x)) + ...
+= f(−x − 2L) + f(x − 4L) + f(−x − 6L) + ...
+=
+∞
+�
+k=1
+f((−1)kx − 2kL)
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx − 2kL) − x0
+σ
+√
+t
+�2�
+5
+
+B
+B'
+B
+0
+XM.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+The proof that fB is a density function is straightforward its definition. Trivially, fB > 0, and by construction:
+� ∞
+−∞
+fB(t)dt =
+� L
+−L
+fB(t)dt =
+� ∞
+−∞
+fX(t)dt = 1
+For practical purposes, we now try to find an upper bound to this expression. Looking at in the model proposed, the
+next constraint is straightforward |(−1)kx − x0| ≤ 2L.
+Following these constraints:
+fB+(x) =
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx + 2kL) − x0
+σ
+√
+t
+�2�
+≤
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2L + 2kL
+σ
+√
+t
+�2�
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�2(k − 1)L
+σ
+√
+t
+�2�
+=
+∞
+�
+k=0
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+� 2kL
+σ
+√
+t
+�2�
+=
+1
+√
+2πσ2t
+∞
+�
+k=0
+exp
+�
+−1
+2
+�4L2
+σ2t
+��k2
+It is known that
+∞
+�
+k=0
+rk2 = 1
+2 + 1
+2ΘE[3, 0, r] where ΘE is the Jacobi theta elliptic function [23]. So:
+fB+(x) ≤
+1
+√
+2πσ2t
+∞
+�
+k=0
+exp
+�
+−1
+2
+�4L2
+σ2t
+��k2
+=
+1
+√
+2πσ2t
+�1
+2 + 1
+2ΘE
+�
+3, 0, exp
+�
+−1
+2
+�4L2
+σ2t
+����
+and
+fB−(x) =
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�((−1)kx − 2kL) − x0
+σ
+√
+t
+�2�
+≤
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2L − 2kL
+σ
+√
+t
+�2�
+=
+∞
+�
+k=1
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2(k + 1)L
+σ
+√
+t
+�2�
+=
+∞
+�
+k=2
+1
+√
+2πσ2t
+exp
+�
+−1
+2
+�−2kL
+σ
+√
+t
+�2�
+=
+1
+√
+2πσ2t
+� ∞
+�
+k=0
+exp
+�
+−1
+2
+�4L2
+σ2t
+��k2
+− 1 − exp
+�
+−1
+2
+�4L2
+σ2t
+���
+=
+1
+√
+2πσ2t
+�1
+2 + 1
+2ΘE
+�
+3, 0, exp
+�
+−1
+2
+�4L2
+σ2t
+���
+− 1 − exp
+�
+−1
+2
+�4L2
+σ2t
+���
+6
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Then,
+fB+(x) + fB−(x) ≤
+1
+√
+2πσ2t
+�
+ΘE
+�
+3, 0, exp
+�
+−1
+2
+�4L2
+σ2t
+���
+− exp
+�
+−1
+2
+�4L2
+σ2t
+���
+= CB
+For each x ∈ [−L, L], fB(x) ≤ fX(x) + CB. Besides CB does not depends on x. Consequently, we have a maximum
+for the density fB which is equal to f(x0) + CB.
+Calling PB = (f(x0) + CB).2ϵ, we can conclude that:
+P(X(t) ∈ (a0 − ϵ, a0 + ϵ)) ≤ PB, for any a0.
+Analogous, CB is the same for the variables Y (t) and Z(t), and we know that f(x0) = f(y0) = f(z0). Then, the
+same result is available for the variables Y (t) and Z(t). According the bounded CB, it is straightforward the uniform
+convergence of the series fB+ and fB− (by the M Weierstrass criteria). An important fact to remark is that PB is not
+even a probability, but in the case in we are interested, we know that is a real number bigger than the probability desired,
+and then, under certain conditions, we can work with it.
+For practical purposes, the error through the CB implementation can be minimized, since the first S terms are available,
+and the tail can be compared with
+S−1
+�
+k=0
+rk2 ≤
+∞
+�
+k=0
+rk2 =
+S−1
+�
+k=0
+rk2 +
+∞
+�
+k=S
+rk2
+And,
+∞
+�
+k=S
+rk2 =
+∞
+�
+k=0
+r(k+S)2 =
+∞
+�
+k=0
+rk2+2kS+S2 = rS2
+∞
+�
+k=0
+rk2r2kS ≤ rS2
+∞
+�
+k=0
+rk2
+Then,
+∞
+�
+k=S
+rk2 ≤ rS2 �1
+2 + 1
+2ΘE[3, 0, r]
+�
+Controlling the value of S controls the value of the error made by truncating the sum. As we said, CB does not depend
+on x, thus, the desired probability can be estimated with any degree of accuracy, according the computational cost
+necessary to this development.
+Taking into consideration the Brownian Motion Theory, in the time lapse of 1 second, the particle position under
+unbounded conditions follows a N(x0, σ2) distribution. To discretize the problem, if we partitioned the time axis of τ
+seconds in τ intervals of 1 second each one, then:
+P(H(t) ∈ Bϵ(a, b, c)) ≤ P(H(t) ∈ Qϵ(a, b, c))
+where Qϵ(a, b, c) denotes the cube centered in (a, b, c) with side size 2 × ϵ. And, considering the independence
+between X(t), Y (t) and Z(t), with X(t) ∈ (a − ϵ, a + ϵ), Y (t) ∈ (b − ϵ, b + ϵ) and Z(t) ∈ (c − ϵ, c + ϵ),
+P(H(t) ∈ Qϵ(a, b, c)) = P(H ∈ Qϵ) is
+P(H ∈ Qϵ) = P(X(t)) × P(Y (t)) × P(Z(t)) ≤ PB × PB × PB = P 3
+B
+For each second τj for τj ∈ {1 : τ}, P(H(t) ∈ Qϵ(a, b, c)) ≤ P 3
+B. Then, under the Wiener process formulation,
+H(τj) ⊥ H(τk|τj) if j ̸= k, j ≤ k.
+7
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+P(H(τj) ∈ Qϵ(a, b, c)) ≤ P 3
+B, ∀τj ∈ {1 : τ}. Calling F :=“# of τj ∈ {1 : T} in which H(τj) ∈ Qϵ(a, b, c)", we are
+interesting in the event F = 0.
+According its nature, F is a Binomial random variable B(τ, P 3
+B). Consequently, the non-collision probability is
+pNC = P(F = 0) ≤ (1 − P 3
+B)τ. At this point, we only can conclude that the probability of the encounter between
+a hydrogen molecule and a Men cluster in a time τ is less than p. We proceed to analyze what happens when the
+number of hydrogen molecules and metallic clusters increase. We emphasize that the H2 molecules have a random
+movement while the clusters are confined in a fixed region of space. Since p is the probability that a random hydrogen
+molecule meets in the cube Qϵ in which a Men cluster is, the most unfavorable case with M clusters is when there is no
+intersection among the cubes that contain it. In this case:
+pA = P(Rτ1 = 0, ..., RτM = 0)
+= 1 −
+M
+�
+i=1
+P(Rτi = 1)
+≥ 1 −
+� M
+�
+i=1
+P(Rτi = 1)
+�
+= 1 − M × p
+In view of this analysis, we can conclude that the non-collision probability is higher than pNC.
+In regular conditions, when this approach is used, the values of pNC and N outcomes into a several numerical instability.
+In this case, the small value of pNC and the large value of N place us in conditions to use the Poisson approach to
+the Binomial distribution (with parameter λ = N × p). Then, P(X = 0) ≈ exp(−λ). Even in the cases when the
+probability is still unavailable, the expected number of collisions is presented according a time window, and then we can
+estimate the probability of collisions in a time window T using the relationship between the Poisson and Exponential
+distributions[26].
+Next, we present the results of the analysis whit different box dimensions (in meters) and number of hydrogen molecules
+(N), according to M = 1.9 × 10
+20 Cu20-clusters [27], where the Cu20-clusters have been considered as spheres.
+3
+Results and analysis
+3.1
+Obtaining non-collision probability values
+The situation we consider is approximately a “realistic”situation, with M = 1.9 × 10
+20 Cu20-clusters in a cubic box
+according to the standard dimensions of reaction chambers (0.1
+3, 0.2
+3 and 0.4
+3 m
+3), and a variable N-H2-molecules
+“contamination”(10
+1 ≤ N ≤ 10
+10). It worked with three temperatures, T, 293, 373 and 473 K. The choice of T is
+arbitrary, conditioned by the possible reaction temperatures [28].
+In Fig. 2 we observe the results obtained for the simulations, considering the maximum sum. That is, take S = 10
+6,
+perform the sum, and add the maximum level for the error. Clearly, a greater probability of non-collision, pNC, is
+observed depending on the increase in volume.
+For a detailed study, we proceed as follows: we model the data obtained through a non-linear graphic fitting considering
+a Boltzmann decrease function, g(x) = A2 +
+A1−A2
+1+exp
+� x−x0
+dx
+� (see Fig. 3). In the Appendix A.2 we show the statistical
+results for each parameter in each data fitting.
+Under these considerations, we can calculate the critical value (criticality)[29, 30] of hydrogen molecules, that is “what
+is the value of N for which the non-collision probability is greater than 1
+2”, i.e. the value of the exponent for which
+1
+2 < pNC.
+It should be clarified that, in the strict physical sense, there is no abrupt phase transition to consider “criticality”. As
+we assumed in the introduction, we consider that there is a chemical reaction if there is an encounter between two
+molecules, and under this assumption we are considering as critical the level of presence of hydrogen for a chemical
+reaction to occur. In any case, it can be demonstrated that there is an “abrupt”transition behavior, for a well defined
+interval in the number of molecules. In Fig. 3 we can observe this behavior.
+8
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Figure 2: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.05m (V1), L = 0.1m (V2) and L = 0.2m
+(V3), at T = 293 K (blue square), 373 K (black star) and 493 K (red triangle).
+Figure 3: Data (blue square) modeling using a non-linear Boltzmann decrease function (green line).
+9
+
+V1
+V2
+1.0 -
+4
+GD
+△
+0.8
+0
+★
+口
+293 K
+0.6
+V
+373 K
+0
+473 K
+文
+0.4 -
+★
+0.2 -
+0.0-
+支立文支安
+123456789101234567891012345678910
+Ln(N)293 K
+1.0 -
+Boltzmann Fit 293 K
+0.8
+0.6
+0.4 
+0.2
+0.0 
+0
+3
+4
+5
+6
+7
+8
+10
+Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Table 1: Critical values obtained from the decrease model for each box and each temperature, for mathematical
+robounds.
+L [m]
+293 K
+373 K
+473 K
+0.05
+3.25
+3.16
+3.09
+0.10
+4.97
+4.86
+4.72
+0.20
+5.96
+5.96
+5.96
+Figure 4: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.05m (V1), at T = 293 K, 373 K and 493
+K. Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical
+rebound model.
+In Table 1 we can see the critical values obtained from the decrease model for each box and each temperature. For the
+smallest volumes, V1 and V2, it is observed that the critical value of N depends more strongly on the temperature than
+in the case of the larger volume (V3). Although it is remarkable the fact of dependence with the size of the box, it can
+be seen directly from Fig. 2. In this way, and under these simplified assumptions, we can obtain control of contaminant
+molecules in relation to the volume and temperature parameters. Linear behavior is evident from the values obtained
+(Table 1, N vs. temperature). Moreover, as the volume increases the slope increases from negative values to null value.
+4
+Conclusion
+By way of conclusion, it can be indicated that considering a Wiener stochastic process, for thermodynamic-statistical
+movements of a gas confined in a box, and considering mathematical rebounds bounded by the physical-geometric
+contour of the problem, the analytical expression could be obtained for the probability density function of encounters
+between two differentiated species of molecules (one of the species fixed in the box -solid or liquid- and the other
+species is a gas whose molecules move stochastically). In addition, the function obtained can be calculated numerically
+or can be bounded. The bounded process allows to reduce the computational cost, and to limit the error from cutting the
+Table 2: Critical values obtained from the decrease model for each box and each temperature, for truncated normal
+model.
+L [m]
+293 K
+373 K
+473 K
+0.05
+3.27
+3.18
+3.12
+0.10
+5.01
+4.91
+4.76
+0.20
+6.00
+5.98
+5.96
+10
+
+293 Trunc
+☆373 Trunc
+473 Trunc
+口
+293 K
+373 K
+473 K
+1.0 0
+0
+★
+Non-collision probability
+0.8
+8
+0.6 -
+0.4
+0.2 +
+0.0+
+口口OO口
+1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910
+Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Figure 5: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.1m (V1), at T = 293 K, 373 K and 493 K.
+Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical
+rebound model.
+Figure 6: Results for the non-collision probability, pNC, vs. ln(N) for L = 0.2m (V1), at T = 293 K, 373 K and 493 K.
+Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical
+rebound model.
+11
+
+293 Trunc
+☆373 Trunc
+→ 473 Trunc
+口
+293
+★373
+473
+★★★
+口
+Non-collision probability
+0.8.
+0.6 -
+0.4 
+0.2 +
+0.0 -
+OOOOG
+1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910
+Ln(N)293 Trunc
+☆373 Trunc
+→ 473 Trunc
+口
+293 K
+373 K
+473K
+1.0-
+Non-collision probability
+0.8
+0.6
+0.4
+0.2 
+0.0-
+123456789101234567891012345678910
+Ln(N)M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+sum in a finite number. In particular, there is an error control that can be made, and it is possible to refine the process
+according to the precision required.
+From the physical-chemical point of view, it is observed that both the number of gas molecules and the dimensions of
+the box affect the probability of encounter. For this model, temperature is a parameter that has a lower incidence on the
+values of the probability of encounter. At this point some considerations have to be made. The first is that in a strict
+sense a chemical reaction is more than the encounter of two chemical entities. The second is the exceptional chemical
+nature of metal clusters, which make them highly reactive. Despite the simplicity of the model we are proposing, this
+model can account in an experiment design about the collision probability between two chemical entities (and this
+collision can lead to a chemical reaction).
+From the point of view of computation, it is a system that requires less computational cost (time + memory) than the
+algorithmic systems developed for this type of problems, so it contributes as a test method in the design of experiments.
+The comparison with an established method (truncated normal model) was optimal. In the method of mathematical
+rebounds the number of molecules needed for a reaction is less than the number obtained by the truncated normal
+model. This is an advantage when strict contamination control is needed.
+On the other hand, in terms of obtaining the density function, mathematical results can be generalized for volumes of
+rectangular prisms of uneven sides. In addition, it remains to calculate the first and second order moments of the density
+function obtained, work that exceeded the purposes of present communication.
+Acknowledgments
+This was was supported in part by PICT-2019-0784, PICT-2017-3944, PICT-2017-1220, PICT-2017-3150 (PICT,
+Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación) and PPID-I231 (PPID,
+Universiad Nacional de La Plata).
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+[25] Charles M Stein. Estimation of the mean of a multivariate normal distribution. The annals of Statistics, pages
+1135–1151, 1981.
+[26] Jeroen Gerritsen and J Rudi Strickler. Encounter probabilities and community structure in zooplankton: a
+mathematical model. Journal of the Fisheries Board of Canada, 34(1):73–82, 1977.
+[27] Leandro Andrini, Germán J Soldano, Marcelo M Mariscal, Félix G Requejo, and Yves Joly. Structure stability of
+free copper nanoclusters: Fsa-dft cu-building and fdm-xanes study. Journal of Electron Spectroscopy and Related
+Phenomena, 235:1–7, 2019.
+[28] Avelino Corma, Patricia Concepción, Mercedes Boronat, María J Sabater, Javier Navas, Miguel José Yacaman,
+Eduardo Larios, Álvaro Posadas, M Arturo López-Quintela, David Buceta, Ernest Mendoza, Gemma Guilera,
+and Álvaro Mayoral. Exceptional oxidation activity with size-controlled supported gold clusters of low atomicity.
+Nature Chemistry, 5(9):775–781, 2013.
+[29] Per Bak and Maya Paczuski. Complexity, contingency, and criticality. Proceedings of the National Academy of
+Sciences, 92(15):6689–6696, 1995.
+[30] Terrie M. Williams. Criticality in stochastic networks. Journal of the Operational Research Society, 43(4):353–357,
+1992.
+Appendix
+A.1
+Errors in the Boltzmann model for the probability calculated according to mathematical rebounds.
+Program used: Origin 9.1
+In all cases, number of points is 10, and degrees of freedon is 6.
+13
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+L=0.05 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.991
+0.009
+A2
+-0.0060
+0.0008
+x0
+4.98
+0.02
+dx
+0.30
+0.03
+Reduced Chi-Sqr 2.66387 × 10
+−4
+Residual Sum of Squares: 0.0016
+Adj. R-Square: 0.99888
+L=0.05 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.981
+0.006
+A2
+-0.005
+0.003
+x0
+3.16
+0.01
+dx
+0.22
+0.02
+Reduced Chi-Sqr 7.93743 × 10
+−5
+Residual Sum of Squares: 4.76246 × 10
+−4
+Adj. R-Square: 0.99957
+L=0.05 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.976
+0.008
+A2
+-0.0011
+0.0009
+x0
+3.10
+0.02
+dx
+0.21
+0.03
+Reduced Chi-Sqr 1.30175 × 10
+−4
+Residual Sum of Squares: 7.8105 × 10
+−4
+Adj. R-Square: 0.99927
+L=0.1 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.991
+0.009
+A2
+-0.0060
+0.0011
+x0
+4.98
+0.02
+dx
+0.30
+0.03
+Reduced Chi-Sqr 2.66387 × 10
+−4
+Residual Sum of Squares: 0.0016
+Adj. R-Square: 0.99888
+L=0.1 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.994
+0.008
+A2
+-0.006
+0.002
+x0
+4.88
+0.02
+dx
+0.33
+0.02
+Reduced Chi-Sqr 1.88888 × 10
+−4
+Residual Sum of Squares: 0.00113
+Adj. R-Square: 0.9992
+14
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+L=0.1 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.996
+0.005
+A2
+-0.0043
+0.0019
+x0
+4.73
+0.01
+dx
+0.33
+0.01
+Reduced Chi-Sqr 7.76599 × 10
+−5
+Residual Sum of Squares: 4.65959 × 10
+−4
+Adj. R-Square: 0.99967
+L=0.2 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.003
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64593 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+L=0.2 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.008
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64587 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+L=0.2 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.008
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64587 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+A.2
+Errors in the Boltzmann model for the probability calculated according to the truncated normal model.
+L=0.05 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.982
+0.006
+A2
+-0.0003
+0.0001
+x0
+3.29
+0.01
+dx
+0.24
+0.01
+Reduced Chi-Sqr 6.6611 × 10
+−5
+Residual Sum of Squares: 0.000399
+15
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Adj. R-Square: 0.99965
+L=0.05 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.982
+0.006
+A2
+-0.004
+0.003
+x0
+3.19
+0.02
+dx
+0.22
+0.02
+Reduced Chi-Sqr 6.69947 × 10
+−5
+Residual Sum of Squares: 4.0196 × 10
+−4
+Adj. R-Square: 0.99960
+L=0.05 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.982
+0.005
+A2
+-0.0004
+0.0003
+x0
+3.19
+0.01
+dx
+0.22
+0.02
+Reduced Chi-Sqr 6.69508 × 10
+−4
+Residual Sum of Squares: 4.01705 × 10
+−4
+Adj. R-Square: 0.99964
+L=0.1 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.991
+0.003
+A2
+-0.0022
+0.0009
+x0
+5.02
+0.07
+dx
+0.21
+0.03
+Reduced Chi-Sqr 5.5392 × 10
+−5
+Residual Sum of Squares: 0.00033
+Adj. R-Square: 0.99977
+L=0.1 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.992
+0.006
+A2
+-0.0047
+0.0025
+x0
+4.92
+0.01
+dx
+0.29
+0.02
+Reduced Chi-Sqr 1.20178 × 10
+−4
+Residual Sum of Squares: 0.000721
+Adj. R-Square: 0.9995
+L=0.1 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.995
+0.005
+A2
+-0.0045
+0.0025
+x0
+4.77
+0.05
+dx
+0.33
+0.01
+Reduced Chi-Sqr 8.70051 × 10
+−5
+Residual Sum of Squares: 5.2203 × 10
+−4
+16
+
+M.L. Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.
+Adj. R-Square: 0.99963
+L=0.2 m, T = 293 K
+Parameter
+Value
+Standard Error
+A1
+0.992
+0.003
+A2
+-0.0078
+0.0065
+x0
+6.01
+0.02
+dx
+0.29
+0.03
+Reduced Chi-Sqr 2.75184 × 10
+−4
+Residual Sum of Squares: 0.00165
+Adj. R-Square: 0.99885
+L=0.2 m, T = 373 K
+Parameter
+Value
+Standard Error
+A1
+0.990
+0.007
+A2
+-0.0068
+0.0075
+x0
+5.99
+0.02
+dx
+0.28
+0.03
+Reduced Chi-Sqr 2.07471 × 10
+−4
+Residual Sum of Squares: 0.00124
+Adj. R-Square: 0.99913
+L=0.2 m, T = 473 K
+Parameter
+Value
+Standard Error
+A1
+0.993
+0.008
+A2
+-0.0082
+0.0025
+x0
+5.97
+0.02
+dx
+0.31
+0.03
+Reduced Chi-Sqr 2.64587 × 10
+−4
+Residual Sum of Squares: 0.00159
+Adj. R-Square: 0.9989
+17
+

9tAzT4oBgHgl3EQfg_wg/content/tmp_files/2301.01476v1.pdf.txt

@@ -0,0 +1,1696 @@
+Lessons Learned Applying Deep Learning Approaches to 
+Forecasting Complex Seasonal Behavior 
+ 
+Andrew T. Karl1, James Wisnowski1, Lambros Petropoulos2 
+ 
+1Adsurgo LLC, Pensacola, FL 
+2USAA, San Antonio, TX 
+ 
+ 
+ 
+Abstract 
+Deep learning methods have gained popularity in recent years through the media and the 
+relative ease of implementation through open source packages such as Keras. We 
+investigate the applicability of popular recurrent neural networks in forecasting call 
+center volumes at a large financial services company. These series are highly complex 
+with seasonal patterns - between hours of the day, day of the week, and time of the year - 
+in addition to autocorrelation between individual observations. Though we investigate the 
+financial services industry, the recommendations for modeling cyclical nonlinear 
+behavior generalize across all sectors. We explore the optimization of parameter settings 
+and convergence criteria for Elman (simple), Long Short-Term Memory (LTSM), and 
+Gated Recurrent Unit (GRU) RNNs from a practical point of view. A designed 
+experiment using actual call center data across many different “skills” (income call 
+streams) compares performance measured by validation error rates of the best observed 
+RNN configurations against other modern and classical forecasting techniques. We 
+summarize the utility of and considerations required for using deep learning methods in 
+forecasting. 
+Key Words: ARIMA, Time Series 
+ 
+ 
+1. Introduction 
+ 
+Member contact call centers receive fluctuating call volumes depending on the day of the 
+week, the time of day, holidays, business conditions, and other factors. It is important for 
+call center managers to have accurate predictions of future call volumes in order to manage 
+staffing levels efficiently. The call center arrival process has been well documented and 
+explored in the literature (Gans, Koole, & Mandelbaum, 2003). In the application presented 
+here, there are several different “skills” (or “splits”) to which an incoming call may be 
+routed – depending on the capabilities of the call center agents – and an arrival volume 
+forecast is required for each skill in the short term for day-ahead or week-ahead predictions. 
+ 
+The weekly seasonality found in call arrivals can be modeled effectively through a variety 
+of methods to include Winter’s Seasonal Smoothing (Winters, 1960) or Autoregressive 
+Integrated Moving Average (Box & Jenkins, 1970). Some accessible references for many 
+of these concepts aimed at the practitioner are well documented in the literature (e.g. 
+Bisgaard & Kulachi, (2007, 2008)) while recommended texts are Bisgaard & Kulachi 
+(2011) and Montgomery, Jennings & Kulachi (2015). 
+ 
+Aiming to improve on these classic methods, “doubly stochastic” linear mixed models 
+(Aldor-Noiman, Feigin, & Mandelbaum, 2009) have effectively modeled additional 
+
+complexities as outlined in a recent review paper from Ibrahim, Ye, L'Ecuyer, & Shen 
+(2016). Similarly, Recurrent Neural Networks (RNNs) have been recommended as deep 
+learning approaches to forecast call volume for a wireless network (Bianchi et al., 2017) in 
+addition to numerous other applications including ride volumes with Uber (Zhu & Laptev, 
+2017). While the doubly stochastic and RNN approaches to predicting call volumes offer 
+greater flexibility in modeling complex arrival behavior by incorporating exogenous 
+variables, this flexibility comes at the cost of greater computational and programming 
+complexity (as well as greater prediction variance). This paper explores practical aspects 
+of managing that complexity for these models, applies the models to actual call volumes 
+recorded by a large financial services company, and compares the prediction capability to 
+that of the more traditional Winters smoothing and ARIMA models.  
+ 
+First, we modify computational aspects of the doubly stochastic approach proposed by 
+Aldor-Noiman, Feigin, & Mandelbaum (2009) to improve call center forecasting 
+performance. Doubly stochastic implies a two-level randomization where not only are call 
+arrivals random variables, but also the call arrival mean parameter. Forecasts are produced 
+by taking advantage of the unique correlation structure for each split while accounting for 
+trend, seasonality, cyclical behavior, and serial dependence. The doubly stochastic model 
+is more complex than ordinary regression as it accounts for both inter- and intra-day 
+correlation. We suggest modifications to the originally proposed approach that lead to more 
+stable convergence and more flexible behavior when many splits need to be fit. 
+ 
+Secondly, we consider how RNNs may be used to model incoming call volume. Whereas 
+“traditional” densely connected, feedforward neural networks process each data point 
+independently, RNNs process sequences according to temporal ordering and retain 
+information from previous points in the sequence. As it processes points within sequences, 
+the RNN maintains states that contain information about what it has seen previously in the 
+sequence (Chollet & Allaire, 2018). This intra-sequence memory is useful in time series 
+applications to autocorrelated data. In the context of call center volumes, these sequences 
+could be constructed to correspond to individual days of observations over a fixed number 
+of (e.g. 30 minute) periods. Bianchi et al. (2017) consider three different RNN architectures 
+to model incoming call volume over a mobile phone network: Elman Recurrent Neural 
+Networks (ERNN) (Elman, 1990), Gated Recurrent Units (GRU) (Cho, et al., 2014), and 
+Long Short Term Memory (LSTM) (Hochrieter & Schmidhuber, 1997), listed in order of 
+increasing complexity.  
+These three RNNs, along with the dense neural network, are now available via the R Keras 
+package (Allaire & Chollet, 2018). Once code has been written for one of the RNNs, the 
+user can switch between the other two by toggling a single option (and, after data 
+reformatting, switch to a dense network). This offers the potential – via a designed 
+experiment – to produce a pragmatic answer to the question of which type of (R)NN 
+provides the best fit to the process at hand. Whereas Bianchi, Maiorino, Kampffmeyer, 
+Rizzi, & Jenssen (2017) created their experimental design by randomly generating points 
+within the design space and then selected the design that lead to the minimum error rate, 
+we create a full factorial design (treating all factors as categorical to allow arbitrary shape 
+in the otherise (discrete) continuous factor of number of nodes) and then explore the 
+behavior of the error rates across the design space with a profiler for the resulitng linear 
+model for the error rate as a function of the NN settings. Unlike ARIMA or regression 
+(including doubly stochastic) modeling approaches for time series, there is a stochastic 
+behavior in the predictions made by neural networks due to the use of randomly initialized 
+weights. Unless the seed for the software’s random number generator is fixed, repeated 
+fitting of the same neural network will lead to different predictions. The amount of 
+
+variation in the resulting predictions depends on the complexity of the network and on the 
+steps that have been taken to avoid overfitting, including early stopping of the optimizer. 
+When selecting a model configuration, we will not only want to minimize the expected 
+error rate, but also minimize the variability in the error rates. To this end, we seek to 
+minimize the upper 95% prediction interval on the testing error rate. The NN study 
+proceeds in two phases where a screening experiment first identifies the most useful 
+(R)NN, followed by a more comprehensive performance study against common 
+forecasting approaches across many more skills.  
+ 
+Section 2 describes the doubly stochastic model for call volumes and how modifications to 
+the originally proposed computational approach can lead to improved convergence. 
+Section 3 details how a full factorial design is used to characterize the performance of RNN 
+options as a function of five factors (and their interactions) on the resulting short-term 
+forecast error rate. Additionally, Section 3 describes the selection of the model 
+configuration that leads to the minimum upper bound on the 95% prediction interval for 
+the testing error rate. Due to the number of different model configurations that must be run 
+along with the computational complexity of RNNs, the first phase discussed in Section 3 
+considers only a limited number of skills and validation days. In Section 4, the best 
+performing RNNs are run over a larger validation set and over all call center skills to 
+compare the performance to the doubly stochastic mixed model approach, and to ARIMA 
+as well as Winters smoothing. 
+ 
+2. Stable Settings for Fitting the Mixed Model 
+ 
+There are two distinct influences on call volumes that induce a correlation between the 
+observed call counts, violating the independence assumption made by ordinary least 
+squares regression models that might be used to model the volumes (Ibrahim & L'Ecuyer, 
+2013). Within a given day, some event may lead to more/fewer calls than expected. For 
+example, unexpected behavior in the stock market in the morning may lead to an increased 
+number of calls for the rest of the day at a financial services contact center. This is intra-
+day correlation. Likewise, there are systemic processes responsible for inter-day 
+correlation. Heuristically, if we noticed that the residuals are very large and positive 
+throughout the day today caused by a weather event for example, we might also expect a 
+larger-than-average call load tomorrow. Ignoring correlation between subsequent 
+observations leads to inaccurate standard errors and prediction intervals. In addition, 
+although the estimates from a linear regression may be unbiased in the presence of 
+correlated residuals, they will not be efficient (Demidenko, 2013). 
+ 
+It is typical for call center regression models to include a day-of-week by period-of-day 
+interaction (Ibrahim, Ye, L'Ecuyer, & Shen, 2016). In a call center open five days per 
+week with 32 half-hour periods per day, this interaction involves 160 fixed-effect 
+parameters. In addition, a call center may require forecasting for holidays. Aldor-
+Noiman, Feigin, & Mandelbaum (2009) exclude holidays when training their model; 
+however, we cannot ignore these days because some splits operate on holidays and may 
+exhibit different behavior on those days. In order to capture this behavior, we include a 
+holiday indicator (holiday_ind) by period-of-day interaction effect in the model. 
+However, some training data sets may include only a single holiday, leading to high 
+variance in the parameter estimates for this effect (each period observation from that one 
+day becomes the new estimate for that period during holidays). To reduce the variability 
+of these estimates, we combine groups of 3 periods together on holidays. That is, periods 
+{1, 2, 3} are assigned p_group = 1, periods {4, 5, 6} are assigned p_group = 2, etc. The 
+
+p_group*holiday_ind interaction is included in the fixed effect structure as an additive 
+effect. 
+ 
+Following Aldor-Noiman et al. (2009), we fit a linear mixed model with correlated errors 
+to the transformed call counts 
+𝑌 = 𝑿𝛽 + 𝒁𝑏 + 𝜀 
+where 
+• 
+𝑌 is the vector transformed call counts, 𝑌 = √𝑐𝑜𝑢𝑛𝑡 + 0.25 
+• 
+𝑿 is a matrix containing the levels of the fixed effects for each observation 
+• 
+𝛽 is the vector of fixed effects parameters containing a day-of-week*period-of-day 
+interaction and a p_group*holiday-indicator interaction 
+• 
+𝒁 is a binary coefficient matrix for the random day-to-day effects in the model. 
+There is one column for each day in the data.  
+• 
+𝑏~𝑁(0, 𝑮 ) is the vector of random day-to-day effects. Each unique day in the data 
+set is represented by one random effect in b. G follows a first-order autoregressive 
+structure, AR(1).  
+• 
+𝜀~𝑁(0, 𝑹 ) is the vector of error terms (residuals), allowing 𝜀 to potentially follow 
+an AR(1) process within days. Thus, R is a block-diagonal matrix, with one AR(1) 
+block for each day in the data set. This accounts for the potential correlation in 
+residuals from proximal periods within days. 
+The full model allows for complex correlation structures. However, for some splits (within 
+particular training data sets), there may be only sporadic and sparse occurrences of call 
+arrivals. This can lead to slow or failed model convergence in some cases. Aldor-Noiman 
+et al. (2009) address this by estimating the doubly stochastic model in two steps: first, the 
+inter-day correlation (G) is estimated using the aggregated total call counts from each day. 
+These parameters are then held constant in a second call to SAS PROC MIXED while 𝛽 
+and 𝑹 are estimated.  
+ 
+Indeed, PROC MIXED can experience convergence problems when the solutions lie on 
+the boundary of the parameter space, such as when variance components are zero (Karl, 
+Yang, & Lohr, 2013).  However, after making modifications to the default PROC MIXED 
+settings, we were reliably able to achieve convergence of the full model with the joint 
+optimization of (𝛽, 𝑮, 𝑹) in a single call to PROC MIXED. In this regard, our approach 
+differs from that of Aldor-Noiman et al. (2009): we fit all of the model parameters jointly 
+(with a single call to PROC MIXED). This will lead to reduced bias in the estimates for 
+the models that do converge.  
+ 
+We improved convergence rates by changing the convergence criterion used by SAS 
+PROC MIXED. By default, SAS ensures that the sum of squared parameter gradients 
+(weighted by the current Hessian of the parameter estimates) is sufficiently small. 
+However, in the presence of strong correlations in the doubly stochastic model, the 
+parameter estimates may lie near the boundary of the parameter space, meaning the 
+gradients may not approach 0 with convergence (Demidenko, 2013). As an alternative, we 
+declare convergence when the relative change in the loglikelihood between iterations is 
+sufficiently small. Additionally, we employ Fisher scoring during the estimation process. 
+Fisher scoring is more stable for models with complex covariance structures and can lead 
+to better estimates of the asymptotic covariance (Demidenko, 2013). Finally, since our 
+
+application only uses the call volume point estimates and not the associated standard errors 
+or tests of significance, we specify ddfm=residual to avoid spending substantial time 
+calculating appropriate degrees of freedom for the approximate F-tests. If confidence or 
+prediction intervals are needed, this value should be set to ddfm=kenwardrodger2 in order 
+to calculate Satterthwaite approximations for the degrees of freedom and to apply the 
+Kenward-Rodger correction (Kenward & Roger, 2009) to the standard errors. The code for 
+our modified approach appears in Figure 1. 
+ 
+Figure 1 Modified SAS code for the Doubly Stochastic Model 
+The square root transformation is applied to reduce the right skew in the observed call 
+volumes, and to stabilize the variance of the observations since quantities such as call 
+volumes tend to follow a Poisson distribution. The approach in Figure 1 employs a normal 
+approximation of this process. We experimented with fitting a mixed Poisson regression to 
+the untransformed call volumes (via PROC GLIMMIX), but found that the run times 
+became unfeasibly long (even when using the default pseudolikelihood approach and 
+avoiding integral approximation) with no noticeable improvement in error rates. 
+ 
+3. Choosing Recurrent Neural Network Configurations with a Designed Experiment 
+ 
+Generally, neural networks consist of layers of weights and nonlinear activation functions 
+that are used to relate inputs (predictors) to outputs (targets). Outputs from each layer are 
+passed sequentially to the next layer as an input vector. The complexity of each layer is 
+determined by the length of the output vector (number of nodes) it produces. A loss 
+function is used to compare the output of the final layer of the neural network to the 
+provided targets (e.g. call volumes), and an optimizer function provides updated values of 
+the weights each node that will decrease the resulting loss. The “depth” of the model is 
+controlled by the number of layers that are used. This “depth” is the source of the phrase 
+“deep learning”. For example, in image processing applications with convolutional neural 
+networks, the different layers can be shown to represent different levels of granularity of 
+detail in an image (Chollet & Allaire, 2018). Besides the number of layers and the 
+number of nodes per layer, there are a number of choices that must be made regarding the 
+properties of the optimizer, the distribution of the random initialization of the parameter 
+weights, and the shape of the activation function(s). 
+ 
+In a traditional, densely connected network, the individual observations are assumed to be 
+independent. A simple example using output from JMP Pro 14.1 helps to illustrate. 
+Suppose we want to fit a densely connected neural network to predict the standardized 
+call count using only the previous day’s standardized call count at the same period (the 
+lag-32 of the call count, since there are 32 periods per day in the example) as a predictor 
+with one node in one layer, using a hyperbolic tangent activation function. This network 
+shown in Figure 2 with resulting weights shown in Figure 3. 
+
+proc mixed data=training_data scoring=50 maxiter=150 maxfunc=10000 convf=1E-6;
+class day_of_week period day_num split p_group;
+by split;
+/* The fixed effects */
+model transf_call_count=day_of_week*period p_group*holiday_ind/
+noint ddfm=residual outp=pred_call_count_output notest;
+/* The day-level random effects */
+/* Note: day num copy is not included in the clAss statment and is numeric *
+random day_num / type=sp(pow)(day_num_copy);
+/* The period-level correlated residuals */
+run; 
+Figure 2 Densely connected neural network with one node in one layer. 
+ 
+Figure 3 Fitted weights from the network with one node in one layer. 
+Suppose that the lag-32 standardized call count is equal to 1 at a given period, t. Then the 
+neural network predicts a standardized call volume of 
+−0.6354 + 3.3923 ∗ TanH(0.5 ∗ (0.4046 + 0.5323 ∗ 1)) = 0.847 
+for the current period, t, where TanH is the hyperbolic tangent function and the 0.5 
+parameter is a fixed value. The nonlinear activation function provides the network with 
+the flexibility to model nonlinear relationships between the inputs and the response, as 
+well as interactions between the inputs. 
+ 
+We next consider a slightly more complex network with two layers using 2 nodes in the 
+first layer and 1 node in the second layer (Figure 4) with parameter estimates shown in 
+Figure 5. 
+ 
+Figure 4 A densely connected neural network with 2 layers using two nodes in the first 
+layer and one node in the second layer. 
+
+Lag[Standardize[cnt_ call], 32]
+Standardize[cnt_call]Estimates
+Parameter
+Estimate
+H1_ 1:Lag[Standardize[cnt_ call], 32]
+0.5323
+H1_1:Intercept
+0.4046
+Standardize[cnt call]_ 1:H1_ 1
+3.3923
+Standardize[cnt call]_2:Intercept
+-0.6354Lag[Standardize[cnt_ call], 32]
+Standardize[cnt_call 
+Figure 5 Fitted weights from the network with two layers. 
+Again assuming that the lag-32 standardized call count is 1, the predicted value is 
+−0.82 + 4.9517
+∗ 𝑇𝑎𝑛𝐻 (0.5
+∗ (0.2416 − 0.7382 ∗ 𝑇𝑎𝑛𝐻(0.5 ∗ (−0.3633 − 0.8785 ∗ 1)) + 0.3057
+∗ 𝑇𝑎𝑛𝐻(0.5 ∗ (−0.0824 + 0.4124 ∗ 1)))) = 0.843 
+In the densely connected network, each observation is processed independently and there 
+is no “memory” of what happened in the previously processed observation. In time series 
+applications, however, there is a temporal ordering that the data are recorded in, and there 
+may be correlation between nearby observations. For example, a spike or drop in call 
+volume might persist over several periods. To address this potential, recurrent neural 
+networks record information when fitting each observation that is then provided as a 
+model input when fitting later observations. 
+ 
+In a simple (Elman) RNN layer, the output from each node (the output of the TanH 
+functions, referred to as the “state”) is recorded and stored, and used as an input for the 
+same node when processing the next observation. Note that there is one state recorded for 
+each node in the layer. For the single layer network example (Figure 3), the state was 
+calculated as 𝑠𝑡 = TanH(0.5 ∗ (0.4046 + 0.5323 ∗ 1)) = 0.44 when the lagged 
+standardized call volume was equal to 1. A simple RNN learns an extra parameter (say, 
+u) to act as a coefficient for the stored state, and the activation function TanH(0.5 ∗
+(𝑤0 + 𝑤1 ∗ 𝑋𝑡)) that is used by the dense network would be replaced by  𝑠𝑡 =
+TanH(0.5 ∗ (𝑤0 + 𝑤1 ∗ 𝑋𝑡 + 𝑢 ∗ 𝑠𝑡−1)) in order to fit a simple RNN. The LSTM and 
+GRU RNNs also use the recorded state when making predictions for the current time 
+period, along with products of additional activation functions that are designed to carry 
+state information further in time. Details of these additional structures are explained in 
+Section 3 of Bianchi et al. (2018). For our purpose it is sufficient to note that the GRU 
+network is extremely similar to the LSTM, albeit less complex due to the ommison of a 
+group of paramters. Chollet & Allaire (2018) remark that Google Translate currently runs 
+using an LSTM with seven large layers. 
+It is not clear a priori which of these four neural networks is most appropriate for a 
+particular call center. Furthermore, it is possible that each of these networks may have a 
+different optimal depth and structure when applied to the call center data. A designed 
+experiment is run to identify the optimal model type and structure. 
+ 
+
+Estimates
+Parameter
+Estimate
+H2_1:Lag[Standardize[cnt _ call], 32]
+-0.8785
+H2_1:lntercept
+-0.3633
+H2_2:Lag[Standardize[cnt_call], 32]
+0.4124
+H2_2:Intercept
+-0.0824
+H1_1:H2_1
+-0.7382
+H1_1:H2 2
+0.3057
+H1_1:lntercept
+0.2416
+Standardize[cnt_ call]_1:H1_1
+4.9517
+Standardize[cnt_ call]_2:Intercept
+-0.82003.1 Data for the Experiment 
+We analyze call volumes aggregated into 30 minute periods from 3 different large-volume 
+skills in an operational call center for the months of March-June 2018. All of the models 
+under consideration are used to forecast next-day call volumes using 5 weeks of training 
+data. Each day contains 32 30-minute periods during which the call center is operating. 
+This application considers the Monday through Friday behavior of the call skills. Due to 
+the use of the one-week lagged observations as a predictor in the neural networks, the first 
+week of training data is not included in the predictor matrix (since the prior week’s call 
+volumes are unknown), meaning each training data set consists of 4*5*32=640 
+observations. The designed experiment in this section will evaluate the methods using a 
+holdout period of the five one-day ahead predictions during last week in June using the 
+three largest skills from the call center. Section 4 will then fit a reduced set of models over 
+all skills for 60 day-ahead predictions. 
+ 
+3.2 Neural Network Input Factors 
+Each network makes use of the one-hot encoding (via a binary indicator matrix) of the day 
+of week and the one-hot encoding of period of day. Furthermore, to capture the day- and 
+week-long correlations, the networks are also fed the call volumes for the same period in 
+the previous day when modeling the current day, as well as the call volumes for the same 
+period on the same day of last week. One-period lagged call volumes are not included, as 
+this is the purpose of the within-sequence memory of the RNNs. Other inputs include a 
+binary indicator for whether the current day is a holiday, a binary indicator for whether 
+yesterday was a holiday, a binary indicator for whether or not last week’s observation was 
+recorded on a holiday, and day number. The day number is a continuous counter for the 
+number of the given day in the data set, which would potentially allow the neural network 
+to detect trends across time. All told, these account for 42 vectors of input for the neural 
+networks. There is no need to create indicator columns for interactions between day of 
+week and period (or any other factors) as the neural network will automatically detect them. 
+ 
+Bianchi et al. (2017) application of hourly call volumes displays strong lag-24 correlation, 
+representing a period-of-day effect. They remove this seasonality by differencing the data 
+at lag-24. By contrast, we do not difference the call volumes, but instead include period-
+of-day (along with day-of-week) as exogenous variables and allow the neural network to 
+detect this seasonality. This approach allows the network to detect the expected interactions 
+between period-of-day and day-of-week, as well as any other input factors. 
+ 
+While these input vectors are included for all models, there are three final input vectors 
+whose (joint) inclusion is treated as an experimental factor: the same-period predictions 
+from the mixed model approach (Aldor-Noiman, Feigin, & Mandelbaum, Workload 
+forecasting for a call center: Methodology and a case study, 2009), from a Winters 
+smoothing model, and from a seasonal ARIMA(1, 0, 1)(0, 1, 1)160 model. The inclusion 
+of the predictions from these models as an input to the neural network is an original 
+approach that gives the network the opportunity to form predictions that may be thought 
+of as corrections to those from these traditional models, based on potential interactions 
+with other included factors. For brevity, we refer to this as the mixed.cheat option, since it 
+allows the neural networks to “cheat” by looking at the predictions generated by these other 
+three models when forming its own predictions for the same time periods. If none of the 
+other 42 input vectors were included with these three, this would represent a supervised 
+learning approach to forming a dynamically weighted average of these three model 
+predictions in order to create a single “bagged” prediction. 
+ 
+
+3.3 Network Configuration Aspects Treated as Experimental Factors 
+The designed experiment considered five different factors of structural settings of the 
+neural networks: model.type {dense, simple (Elman) RNN, GRU, LSTM}, nlayers {1, 
+2}, nnodes (per layer) {25, 50, 75, 100}, kernel.L2.reg {0, 0.0001} and mixed.cheat 
+{FALSE, TRUE}. The L2 regularizer adds kernel.L2.reg times each weight coefficient to 
+the total loss function for the network. Similar to Lasso regression, this helps bound the 
+magnitude of the model coefficients and could potentially help prevent overfitting the 
+training data.  
+ 
+We fit a full factorial design (requiring 128 runs) for these five factors, which allows us 
+to test for the presence of up to five-way interactions between the five factors. Figure 6 
+shows the first 6 runs of the design. The design is replicated over the 3 largest splits and 
+across the 5 subsequent one-day ahead forecasts. This produces a total of 1920 runs for 
+the entire experiment. The replication allows for behavior to be averaged over different 
+days and for a more detailed exploration of how the variance of the error rates depends on 
+each factor. 
+ 
+Figure 6 First 6 runs of the designed experiment 
+3.4 Static Considerations for Neural Network Configuration 
+The input for the classical dense neural network is a 640x42 matrix (640x45 if 
+mixed.cheat=TRUE). The dense network does not consider the temporal ordering of the 
+640 observations (outside of the explicit inclusion of the day- and week-lagged 
+observations as inputs): the observations are shuffled after each epoch and then processed 
+in batches (we used a batch size of 32). 
+ 
+By contrast, the RNNs consider the ordering of the observations. The input for the RNNs 
+is a 20x32x42 array. This indicates to the RNN that there are 20 batches (days) of 32 
+timesteps (periods) with 42 predictors per time step. By default, the batches are treated 
+independently and the timesteps within each batch are potentially correlated (via the 
+persistence of the states in the RNN). If the batches themselves are presented in a 
+temporal order (as is the case in our application), then this can be indicated to the Keras 
+model via the STATEFUL=TRUE option and by disabling the shuffling of batches 
+during training. This retains the model weights from batch-to-batch (day-to-day) to allow 
+for possible long-term behavior. However, we found that using the STATEFUL option 
+led to a failure to converge in some skill*day combinations and the resulting validation 
+error rates were not significantly different from those generated without the STATEFUL 
+option. This seems to indicate that the dependence on prior days’ behavior is already 
+captured by the inclusion of the one-day and one-week lagged observations. Due to the 
+occasional convergence issues, the results in the sequel are generated with 
+STATEFUL=FALSE (and batch shuffling enabled during training). It would have also 
+been possible to fit the model with week- or month-long batches by training the RNN on 
+a 4x160x42 or a 1x640x42 array. We did not consider the week-long batches, but the 
+
+model.type
+nlayers mixed.cheat nnodes
+kernel.L2.reg
+layer_simple_rnn
+2 
+FALSE
+50
+0.0001
+layer_ gru
+FALSE
+25
+0
+layer_Istm
+1
+TRUE
+100
+0
+layer_gru
+TRUE
+75
+0.0001
+layer_gru
+TRUE
+75
+0
+layer_simple_rnn
+FALSE
+100
+0month-long batches tended to produce inferior predictions to the day-long batches. This 
+could possibly be due to the lack of long term correlations in the data and the fact that the 
+smaller batches allow for the shuffling of the ordering that the days are fed through the 
+gradient-optimization routine of the neural network, which can improve the model fit by 
+preventing the model from overweighting the first observations that are provided to the 
+network in each epoch (Chollet & Allaire, 2018). 
+ 
+Note that the model weights are reset and the model is retrained for each skill. This is in 
+contrast to the approach taken by Zhu & Laptev (2017) in which a single network is fit to 
+accommodate disparate behavior from different cities. As discussed in Section 5, a single 
+multi-output network could potentially be built to model all of the skills at once. 
+ 
+While they were not included as experimental factors in this application, we also noticed 
+a significant relationship between the quality of the predictions and the optimization 
+routine employed. Extensive pilot experimentation led to our use of the AMSGrad variant 
+(Reddi, Kale, & Kumar, 2018) of the Adam optimizer (Kingma & Ba, 2014), with a 
+learning rate decay of 0.0001. We would recommend including both the optimizer and 
+the optional learning rate decay as factors in the designed experiment for parameter 
+tuning in future problems. 
+ 
+We found it was important to tune the number of epochs for each model fit using 
+validation data (the last week in the training set) by first fitting 500 epochs for each 
+model, taking a moving average (with a window size of 10 epochs) of the resulting 
+WAPEs on the validation data, and then refitting the model (on both the training and the 
+validation data in order to predict an additional day which was held out as a test set) with 
+the number of epochs that produced the minimum validation WAPE. The moving 
+average is important due to the volatile and non-monotonic behavior we observed in the 
+individual recorded WAPEs and helps to find a relatively stable region. Consistent with 
+previous findings (Bianchi, Maiorino, Kampffmeyer, Rizzi, & Jenssen, 2017), we noticed 
+that the RNNs take many more epochs to converge than the dense neural network. Other 
+researchers (such as Bianchi et al.) have used more than 500 epochs when fitting RNNs, 
+so this upper bound should also be considered as an important factor when building an 
+RNN.  
+ 
+While we also experimented with recurrent dropout (Gal & Ghahramani, 2015) to 
+prevent overfitting, it led to degraded performance in the early iterations of our 
+experiment and we removed it from consideration. However, this could simply be due to 
+features of this particular application, such as the relatively short training period of five 
+weeks: Chollet & Allaire  (2018) strongly advocate the use of dropout and recurrent 
+dropout. 
+ 
+The models experienced improved errror rates after switching the kernel initializer for the 
+random weights to the He normal initializer (He, Zhang, Ren, & Sun, 2015). We used the 
+relu activation function exclusively, although this choice could also impact the quality of 
+the resulting model fit. And while our application did not detect any two-factor 
+interactions among the five experimental factors, it is possible that some of these 
+additional factors could depend on the type of (R)NN being used, meaning there would 
+be an interaction between these factors and model.type.  
+ 
+A final contributing factor is the batch size. This determines how many observations are 
+processed before the gradients are updated: when fitting Keras models on a GPU, amount 
+
+of memory available on the GPU can be a limiting factor on the batch size. This choice is 
+more constrained in the RNNs, where each batch is a single day/week/month (determined 
+by the number of timesteps specified in the input array to the RNN). By contrast, the 
+batch size for a dense network can be set between 1 and the number of observations in 
+the training data. We noticed significant differences in the error rates from the dense 
+model depending on what batch size was used. 
+ 
+3.5 Experimental Response 
+While mean squared error (MSE) is frequently used to evaluate and compare predictive 
+models, this is a poor metric for the call center application as it will give undue focus to 
+the low call volume periods at the beginning and end of each day. Instead, the weighted 
+absolute percentage error (WAPE) is recommended for call volume modeling (Ibrahim, 
+Ye, L'Ecuyer, & Shen, 2016). This weights the absolute percentage error in each period 
+by the number of calls received in that period and is defined by 
+𝑊𝐴𝑃𝐸 =
+∑
+|𝑌𝑖 − 𝑌̂𝑖|
+𝑛
+𝑖=1
+∑
+𝑌𝑖
+𝑛
+𝑖=1
+ 
+where 𝑌𝑖 and 𝑌̂𝑖 are the observed and predicted volumes, respectively, for each 30 minute 
+period 𝑖 = 1, … , 𝑛. Because WAPE is our metric of interest, the models are compiled to 
+use a mean absolute error loss-function, which minimizes the numerator of the WAPE 
+(the denominator is static).  
+ 
+For each row of the experimental table (Figure 6), the specified neural network is fit 
+(independently) to model five subsequent days of each split. That is, day 1 is predicted 
+using the previous 5 weeks leading up to day 1, then the model is reset and day 2 is 
+predicted using the 5 weeks leading up to day 2, etc.   The vector of five one-day ahead 
+predictions is compared with the observed call volumes (that were not visible to the 
+model during training), and the resulting WAPE is recorded. 
+ 
+3.6 Analysis of Experimental Results 
+Figure 7 gives a typical output of the prediction error across the different formulations of 
+the neural networks. GRU often has the lowest forecast error or at least is consistently 
+close to the lowest. The other procedures tend to have much more unstable performance 
+based on the choice of nodes and layers as well as across the days and splits. 
+ 
+Figure 7 Forecast error by model type and settings for forecast day 5 split 3 
+Regression analysis is used to determine the statistically significant factors and 
+interactions. In order to control a heavy right-skew in the recorded WAPEs, an inverse 
+
+model.type
+NN Classic
+RNN GRU
+ RNN LSTM RNN Simple
+WAPE
+WAPE
+WAPE
+WAPE
+nlayers
+nnodes
+Mean
+Mean
+Mean
+Mean
+1
+25
+6.7%
+6.2%
+8.5%
+6.0%
+50
+7.1%
+6.2%
+6.5%
+6.7%
+75
+7.1%
+5.8%
+6.8%
+5.9%
+100
+7.0%
+6.4%
+6.9%
+6.8%
+2
+25
+7.5%
+6.6%
+8.1%
+6.9%
+50
+7.3%
+5.8%
+6.9%
+7.0%
+75
+7.1%
+6.0%
+6.6%
+6.5%
+100
+8.0%
+5.9%
+7.5%
+6.0%transformation is applied to use as the response in the regression models. Figure 12 
+displays the original skewness and the transformation to normality after taking the 
+inverse of the WAPEs.  
+ 
+3.7 Loglinear Variance Regression Model 
+The mean structure of 1/WAPE is modeled by including the main effects of the five 
+experimental factors (model.type, nlayers, mixed.cheat, nnodes, kernel.L2.reg) and all of 
+the interactions. In addition, split, file (which represents the validation day), and split*file 
+are included as blocking factors.   
+ 
+While an ordinary regression model for the full factorial design would assume the same 
+error variance for all responses across the design space, the loglinear variance model 
+allows for the error variance itself to be modelled as a function of the input factors. This 
+is appealing for this application, where parsimonious networks may be expected to 
+demonstrate less variability during repeated fittings.  
+ 
+For each factor setting, the loglinear variance model produces a predicted mean of 
+1/WAPE and a predicted standard deviation of 1/WAPE, representing the variability due 
+to the random initial weighting of the NN. 
+We include model.type, nlayers, mixed.cheat, nnodes, kernel.L2.reg, file, and split as 
+factors in the loglinear variance model (using JMP Pro 14.1), which models the log of the 
+error variance as a linear combination of these factors (which are all treated as 
+categorical). With the exception of kernel.L2.reg, all of these effects are found to be 
+significantly associated with the error variance (Figure 8). 
+ 
+Figure 8 Factors with a significant impact on WAPE variability 
+For the mean model, model.type, nlayers, mixed.cheat, nnodes, day, split, and file*split 
+were found to be significantly associated with 1/WAPE (Figure 9). Neither kernel.L2.reg 
+nor any of the interaction terms involving the NN architecture were found to be 
+significant.  
+ 
+ 
+
+Variance Effect Likelihood Ratio Tests
+Source
+Test Type
+DF
+ ChiSquare
+Prob>ChiSq
+model.type
+Likelihood
+3
+114.126
+<.0001*
+nlayers
+Likelihood
+1
+4.6333
+0.0314*
+mixed.cheat
+Likelihood
+1
+53.6884
+<.0001*
+nnodes
+Likelihood
+3
+16.5624
+0.0009*
+kernel.L2.reg 
+Likelihood
+1
+0.0042
+0.9482
+file
+Likelihood
+4
+36.8213
+<.0001*
+split
+Likelihood
+2
+56.3123
+<.0001* 
+Figure 9 Factors that are significantly associated with the mean WAPE 
+Removing the insignificant interaction terms (but allowing kernel.L2.reg to remain in 
+both the mean and variance models) produces the profiler shown in Figure 10. Notice the 
+large standard deviation associated with the LSTM model. The dependence on file and 
+split is not shown, since there are no interactions modeled between these and the other 
+experimental factors. While we want to maximize 1/WAPE, we also want to minimize 
+the standard deviation of 1/WAPE. That is, we would like to maximize the lower 95% 
+prediction interval of 1/WAPE (or minimize the upper 95% PI of WAPE) in order to find 
+the model configuration that will give a combination of relatively good results on average 
+while being protected against large errors. 
+
+Fixed Effect Tests
+Source
+Nparm
+DF
+DFDen
+F Ratio
+Prob > F
+model.type
+m
+m
+796.5
+44.6120
+<.0001*
+nlayers
+1
+1
+1258
+5.5213
+0.0189*
+model.type*nlayers
+3
+m
+796.5
+1.0691
+0.3614
+mixed.cheat
+1
+1
+1258
+17.0652
+<.0001*
+model.type*mixed.cheat
+3
+3
+796.5
+1.3181
+0.2672
+nlayers*mixed.cheat
+1
+1
+1258
+2.3413
+0.1262
+model.type*nlayers*mixed.cheat
+3
+3
+796.5
+1.9528
+0.1196
+nnodes
+m
+786.2
+4.5600
+0.0036*
+model.type*nnodes
+9
+5
+839.4
+0.7864
+0.6290
+nlayers*nnodes
+3
+3
+786.2
+0.7371
+0.5301
+model.type*nlayers*nnodes
+9
+6
+8394
+0.6668
+0.7395
+mixed.cheat*nnodes
+3
+3
+786.2
+0.6624
+0.5753
+model.type*mixed.cheat*nnodes
+9
+5
+839.4
+0.2224
+0.9913
+nlayers*mixed.cheat*nnodes
+m
+786.2
+0.9291
+0.4261
+model.type*nlayers*mixed.cheat*nnodes
+9
+9
+839.4
+0.4312
+0.9186
+kernel.L2.reg
+1
+1
+1258
+0.0026
+0.9590
+model.type*kernel.L2.reg
+m
+796.5
+0.1263
+0.9445
+nlayers'kernel.L2.reg
+1
+1
+1258
+0.2313
+0.6306
+model.type*nlayers*kernel.L2.reg
+3
+m
+796.5
+0.7812
+0.5046
+mixed.cheat*kernel.L2.reg
+1
+1
+1258
+0.0295
+0.8636
+model.type*mixed.cheat*kernel.L2.reg
+3
+m
+796.5
+0.1395
+0.9364
+nlayers*mixed.cheat*kernel.L2.reg
+1
+1
+1258
+0.1498
+0.6988
+model.type*nlayers*mixed.cheat*kernel.L2.reg
+3
+3
+796.5
+0.3776
+0.7692
+nnodes*kernel.L2.reg
+m
+m
+786.2
+0.9118
+0.4347
+model.type*nnodes*kemel.L2.reg
+9
+6
+839.4
+0.8905
+0.5331
+nlayers*nnodes*kernel.L2.reg
+3
+m
+786.2
+0.7442
+0.5259
+model.type*nlayers*nnodes*kernel.L2.reg
+9
+839.4
+0.6535
+0.7513
+mixed.cheat*nnodes*kernel.L2.reg
+3
+3
+786.2
+0.0812
+0.9703
+model.type*mixed.cheat*nnodes*kernel.L2.reg
+9
+8394
+0.2691
+0.9827
+nlayers*mixed.cheat*nnodes*kernel.L2.reg
+3
+m
+786.2
+0.8510
+0.4662
+model.type*nlayers*mixed.cheat*nnodes*kernel.L2.reg
+9
+6
+8394
+0.6614
+0.7442
+file
+4
+4
+668.1
+1258.316
+<.0001*
+ds
+2
+2
+857.3
+1867.911
+<.0001*
+file*split
+8
+8
+746.8
+244.5502
+<.0001* 
+Figure 10 Mean and Standard Deviation of 1/WAPE as a function of the experimental 
+factors. Goal is to maximize the top (1/mean forecast error) and minimize the bottom 
+(standard deviation) 
+Figure 11 plots the upper 95% prediction interval for WAPE against the experimental 
+factors. The model configuration that minimizes the upper 95% PI of WAPE is a GRU 
+model that is allowed to use the MIXED, ARIMA, and Winters predictions as inputs, 
+with 2 layers and 50 nodes per layer and an L2 penalty of 0.0001 on the kernel weights.  
+However, the L2 penalty was not found to be significant in either the mean or the 
+variance models and the contribution of nlayers is relatively flat.  
+ 
+Figure 11 Upper 95% Prediction Interval for WAPE 
+To summarize, the designed experiment provided insight to effectively select the 
+appropriate levels across several neural network models and parameters. The goal is to 
+balance model performance in minimizing forecast error (that is, maximizing 1/forecast 
+error) and minimizing the variance of this forecast error. Figure 10 and Figure 11 are 
+interpretable across all factors and settings as displayed due to the absence of significant 
+interactions between the factors. The top half of Figure 10 displays the reciprocal forecast 
+error (the larger the value the better) which indicates preferred settings of GRU or Elman 
+(simple) for the model, 1 layer, using the mixed model forecasts, and 25 nodes while the 
+selection of L2.kernel makes little difference. The lower half of Figure 10 displays the 
+variance (lower is better) where the traditional neural net would be preferred along with 2 
+
+Prediction Profiler
+30 -
+28.87292
+29
+[27.9033,
+28
+29.8426]
+27
+26
+Dev
+2.5
+ [1.66636,
+1.5
+dense
+layer_gru
+layer_Istm
+2
+FALSE-
+TRUE-
+5
+0.0001
+layer_gru
+2
+TRUE
+50
+0.0001
+model.type
+nlayers
+mixed.cheat
+nnodes
+kemel.L2.regPrediction Profiler
+0.044
+0.0435
+0.043
+0.040571
+0.0425
+0.042
+0.0415
+0.041
+0.0405
+dense
+layer_gru
+layer_Istm
+layer_simple_rnn
+2
+FALSE-
+TRUE-
+5
+9
+5
+100-
+0.0001
+layer_gru
+TRUE
+50
+0.0001
+model.type
+nlayers
+mixed.cheat
+nnodes
+kenel.L2.reglayers, using mixed model forecasts, with 50 nodes while robust to the regularization 
+parameter value. Note that the traditional dense neural network does have consistently 
+worse forecast error across all scenarios and could not be recommended despite having 
+the lowest variance. The prediction interval on forecast error is an alternative view that is 
+preferred by many practitioners where the goal is to minimize the width. Figure 11 (lower 
+is better) clearly shows GRU is the preferred solution using the mixed model forecasts 
+with 50 nodes.  
+ 
+ 
+Figure 12 Representative distribution of WAPE and 1/WAPE resulting from the designed 
+experiment 
+ 
+4. Comprehensive Performance Study 
+ 
+Based on the results of the designed experiment for NN error rates, we perform a further 
+study with actual call center data across 36 skills rather than only 3. We consider a 
+consistent 5-week training period advancing across multiple months of data to produce 60 
+one-day ahead predictions. These predictions do not use the actual data for the forecasted 
+day during model training and can be viewed as a validation set for the trained models. 
+This allows us to compare the relative performance of the mixed model and the neural 
+networks. We also include the error rates from the ARIMA and Winters seasonal 
+smoothing models for comparison. 
+ 
+Though we could have simply chosen the GRU RNN with 50 nodes on each of 2 layers 
+with kernel.L2.reg complexity parameter set to 0.0001 as the only representative RNN 
+based on the designed experiment, we decided to include all 4 neural network methods 
+
+Distributions
+WAPE
+0 0.1
+0.3
+0.5
+0.7
+0.9 1 1.1
+1.3
+1.5
+1/WAPE
+0
+5
+10
+15
+20
+25 
+30screened by using these same parameter settings. It is possible with the added skills – 
+with many having much lower call volumes – that another RNN model could work better 
+than GRU.  
+ 
+4.1 Results: One-day ahead Predictions Over 60 Separate Validation Days 
+Figure 13 provides the forecast errors averaged across all 60 days for the 12 splits with 
+the highest call volume sorted in descending call volume order. Note that results in this 
+section are generated by the (R)NNs with the mixed.cheat option disabled. The Doubly 
+Stochastic Mixed Model has the lowest average forecast error for all but Split 5540 and is 
+often significantly lower than the competitors. Winters Seasonal Exponential Smoothing 
+also performs quite well given its relative simplicity. The GRU performance confirms the 
+results from the designed experiment as usually the best RNN and always competitive 
+with the best. The highly complex LSTM recurrent neural network has very large forecast 
+errors for many of these splits and cannot be recommended. Note also that forecast error 
+generally increases for all methods as call volume decreases.   
+ 
+Figure 13 Average WAPE forecast errors across 60 separate validation days for high 
+call-volume splits 
+Figure 14 shows the similar trend of increasing error rates with decreasing call volumes 
+for the medium call volume splits. The GRU recurrent neural network is usually 
+outperforming the other neural network methods and is closer to the error rates of the 
+Doubly Stochastic Mixed Model. 
+ 
+Figure 14 Average WAPE forecast errors across 60 separate validation days for medium 
+call-volume splits 
+For the low call volume splits displayed in Figure 15, the GRU and Simple recurrent 
+neural networks perform similarly and slightly better than Doubly Stochastic and Winters 
+for most of the splits. The very low call volumes (last 3 splits) do seem to benefit from 
+the recurrent neural network formulation. 
+
+Split
+Sum Call Vol
+ARIMA
+Doubly Stoch
+NN_Classic
+RNN_GRU
+RNN_LSTM
+RNN_Simple
+Winters
+5000&5240
+929754
+9.0%
+8.4%
+10.2%
+11.2%
+38.0%
+13.9%
+8.3%
+5400&5570
+461256
+7.4%
+6.6%
+8.8%
+8.4%
+26.1%
+14.3%
+7.1%
+5620
+162996
+8.6%
+7.7%
+9.6%
+9.1%
+28.7%
+9.6%
+8.4%
+5660
+137759
+10.2%
+10.1%
+13.2%
+12.0%
+17.1%
+14.0%
+10.5%
+5020
+71930
+11.5%
+11.2%
+14.5%
+12.3%
+27.3%
+14.2%
+11.7%
+5630
+65079
+15.4%
+14.9%
+16.7%
+15.5%
+26.4%
+15.5%
+14.1%
+5840
+63984
+13.1%
+12.1%
+14.6%
+13.4%
+25.3%
+12.9%
+13.0%
+5200
+48728
+13.2%
+13.4%
+15.6%
+13.5%
+52.4%
+13.5%
+13.6%
+5540
+38236
+15.9%
+15.8%
+16.8%
+16.3%
+28.5%
+16.6%
+16.0%
+5670
+34793
+14.0%
+13.2%
+15.6%
+14.2%
+27.3%
+13.9%
+14.0%
+6500
+30534
+16.8%
+16.4%
+18.5%
+17.3%
+23.0%
+18.0%
+17.1%
+5260
+23849
+21.9%
+20.9%
+22.7%
+20.3%
+32.5%
+20.2%
+21.2%Split
+Sum Call Vol
+ARIMA
+Doubly Stoch
+NNClassic
+RNN_GRU
+RNN_LSTM
+RNN_Simple
+Winters
+5460
+20922
+18.2%
+18.1%
+19.5%
+18.1%
+26.0%
+17.7%
+18.2%
+5410
+19461
+19.8%
+19.2%
+21.4%
+20.0%
+59.1%
+20.8%
+19.7%
+6350
+16874
+30.8%
+26.8%
+29.3%
+28.7%
+31.1%
+29.0%
+29.4%
+5060
+16765
+19.3%
+19.1%
+20.7%
+19.6%
+24.4%
+19.6%
+19.6%
+5650
+9911
+23.8%
+23.4%
+24.6%
+24.0%
+27.3%
+24.3%
+24.0%
+5030
+8102
+40.0%
+26.0%
+28.5%
+27.8%
+33.2%
+27.7%
+26.9%
+5680
+7525
+27.3%
+27.1%
+28.2%
+26.7%
+31.9%
+27.8%
+27.7%
+5440
+7402
+28.5%
+28.1%
+30.3%
+28.3%
+33.8%
+29.7%
+29.2%
+5070
+6446
+34.6%
+34.5%
+34.6%
+33.5%
+43.6%
+35.0%
+34.7%
+5420
+5247
+35.0%
+34.6%
+36.2%
+33.9%
+38.1%
+34.7%
+35.0%
+5899
+4844
+36.4%
+35.8%
+37.0%
+35.2%
+64.8%
+36.6%
+36.7%
+5100
+4019
+34.7%
+34.0%
+36.6%
+34.3%
+40.1%
+35.5%
+34.8% 
+Figure 15 Average WAPE forecast errors across 60 separate validation days for low 
+call-volume splits 
+Overall, the best performing procedures were the Doubly Stochastic Mixed Model and 
+the GRU Recurrent Neural Network. Figure 16 shows forecast error by split sorted by 
+call volume. Generally, the mixed model does better for large and medium volume splits 
+(lower is better on the graph) while the RNN 
+model is more effective for the small volume—particularly the very small volume splits. 
+  
+ 
+Figure 16 Forecast error rates ordered by call volume by split for Doubly Stochastic 
+(blue) and GRU (red) 
+Figure 17 presents a different view of this same pattern. For each of the 60 one-day ahead 
+forecasts within each split, the GRU and Doubly Stochastic WAPEs are recorded, along 
+with the number of calls recorded for that split over the training and validation data 
+(sum_all_calls). For each split, Figure 17 plots the percent of the 60 day-ahead forecasts 
+for which GRU “won” (GRU WAPE < Doubly Stochastic WAPE) against the log of the 
+median call volume recorded by that split over the 60 different pairs of training and 
+validation data. There appears to be a linear decrease of the relative performance of GRU 
+against Doubly Stochastic in the log of the call volume. 
+
+Split
+Sum Call Vol
+ARIMA
+Doubly Stoch
+NN_Classic
+RNN_GRU
+RNN_LSTM
+RNN_Simple
+Winters
+5710
+3742
+39.6%
+38.8%
+40.7%
+39.1%
+45.7%
+40.2%
+38.9%
+5820
+2949
+44.1%
+43.1%
+46.1%
+43.2%
+50.4%
+44.0%
+43.4%
+5690
+2238
+56.1%
+51.1%
+51.6%
+51.2%
+58.9%
+51.8%
+52.5%
+5220
+2089
+49.7%
+49.5%
+50.5%
+49.3%
+54.0%
+50.1%
+49.7%
+5470
+1975
+49.2%
+48.8%
+52.0%
+50.5%
+57.0%
+50.7%
+50.2%
+6330
+1556
+60.5%
+60.1%
+61.0%
+60.2%
+65.1%
+61.6%
+60.7%
+5040
+1398
+79.0%
+69.0%
+69.8%
+73.4%
+72.2%
+71.1%
+80.7%
+6310
+922
+98.5%
+92.0%
+92.2%
+88.1%
+91.0%
+89.5%
+92.8%
+5720
+918
+80.2%
+77.8%
+77.5%
+77.6%
+85.0%
+78.4%
+80.1%
+6370
+485
+95.3%
+93.6%
+96.5%
+91.3%
+111.3%
+94.0%
+95.2%
+6360
+171
+150.4%
+136.8%
+126.5%
+107.7%
+118.7%
+111.7%
+142.5%
+6340
+14
+189.7%
+161.4%
+188.2%
+102.7%
+108.9%
+107.8%
+187.7%Forecast ErrorbySplit AverageOver 6oDays
+1.6 
+IDoubly Stoch
+1.5 
+RNN GRU
+1.4
+1.3 
+1.2
+1.1 -
+Errot
+aber
+1.0 
+0.9
+0.8
+0.7
+pa
+ubiay
+0.6
+0.5 
+0.4 
+0.3 
+0.2 
+0.1
+HighCallVolum
+Medium CallVolume
+Low Call Volume 
+Figure 17 Percent of the 60 day-ahead forecasts for which the GRU WAPE was less than 
+the Doubly Stochastic WAPE for each of the 36 splits plotted against the log of the 
+median sum of all calls recorded for each split over each pair of training and validation 
+data. 
+For the large call-volume splits, the extra flexibility of the GRU model does not lead to 
+improvements over the predictions generated by the mixed model approach. Because the 
+mixed model computations are faster and the implementation is less complex, there does 
+not appear to be any benefit to running the neural networks for the high-volume splits for 
+this short-term application. This is consistent with the findings of the Uber traffic volume 
+study when short-term predictions were considered (Zhu & Laptev, 2017), and would 
+possibly change if a longer training period (several months or multiple years) were used 
+for the call center data.  
+ 
+4.2 Improving GRU RNN by Using Doubly Stochastic Forecasts as a Covariate 
+Based on pilot studies during the designed experiment, the forecasting performance of the 
+GRU recurrent neural network often improved by integrating the forecasted value for the 
+validation days from the doubly stochastic, ARIMA, and Winters models. This 
+“cheating” by using other models’ forecasts (shown as mixed.cheat) proved to be a 
+significant benefit for the GRU model over these 60 predictions for each skill. The 
+WAPE for the held-out validation data was again the primary measure of performance. 
+ 
+Figure 18 displays the forecast errors for the high call volume skills averaged over the 60 
+validation days for the each of the neural networks when they use the mixed forecasts. 
+Note the side-by-side comparison of the RNN_GRU (no cheat) and GRU_cheat columns 
+where in most cases the “cheating” does result in improved forecasts and in those cases 
+where it is not better, it has only marginally declined. Additionally, the Simple_cheat 
+error rates compare favorably with the GRU_cheat while the LSTM_cheat suffers from 
+significantly poorer performance and instability issues. The doubly stochastic forecast is 
+still quite good and often the best choice. These results are also consistent when looking 
+at the medium and low call volume splits. Therefore, we recommend using the forecasts 
+from a doubly stochastic or Winters model as inputs to recurrent neural networks.  
+
+Percent won by GRU vs. Log[Median(sum all calls)]
+100%
+ Percent won by GRU
+%06
+%08
+70%
+Percent won by GRU
+%09
+50%
+40%
+30%
+20%
+10%
+0%
+4
+6
+8
+10
+12
+14
+Log[Median(sum all_ calls)] 
+Figure 18 Forecast errors for high call volume splits averaged over 60 validation 
+forecast days allowing NN to “cheat” to improve predictions 
+The median of the GRU WAPE (mixed.cheat=FALSE) minus the GRU WAPE 
+(mixed.cheat=TRUE) (within each split/day combination, for a sample size of 
+36*60=2160) is 0.002 with a p-value of 0.0005 from the Wilcoxon signed rank test with a 
+null hypothesis that the differences were drawn from a population with median equal to 
+0, indicating that mixed.cheat does tend to improve the performance of the GRU model. 
+ 
+A similar comparison of paired differences of error rates (all with mixed.cheat=FALSE) 
+confirms that GRU outperforms the other (R)NNs: LSTM - GRU produces a median of 
+0.0247 with a p-value of 1e-129, NN Classic - GRU produces a median of 0.0690 with a 
+p-value of <1e-185, and Simple RNN - GRU produces a median of 0.0021 with a p-value 
+of 1e-04. 
+ 
+ 
+References 
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+Split
+Sum Call Vol
+Doubly Stoch
+RNN_GRU
+GRU_cheat
+LSTM_cheat
+Simple_cheat
+5000&5240
+926493
+6.7%
+10.2%
+7.8%
+14.4%
+7.8%
+5400&5570
+459961
+6.3%
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+7.4%
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+164138
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+71900
+11.2%
+12.4%
+12.4%
+27.0%
+12.3%
+5630
+65279
+11.5%
+13.5%
+13.7%
+421.5%
+13.2%
+5840
+63687
+12.0%
+13.2%
+13.1%
+26.1%
+12.7%
+5200
+48624
+13.0%
+14.2%
+14.4%
+28.1%
+14.0%
+5540
+38318
+17.1%
+16.3%
+16.9%
+20.8%
+17.5%
+5670
+34741
+13.4%
+14.7%
+14.4%
+24.5%
+14.2%
+6500
+30352
+16.3%
+17.5%
+16.8%
+35.2%
+17.2%
+5260
+23674
+20.7%
+21.3%
+20.5%
+28.0%
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+Recurrent Neural Networks. arXiv. Retrieved from 
+https://arxiv.org/abs/1512.05287 
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+and research prospects. Manufacturing and Service Operations Management, 
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+Forecasting with Neural Networks at Uber. ICML 2017 Time Series Workshop. 
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+arXiv. Retrieved from https://arxiv.org/abs/1709.01907 
+ 
+ 
+

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+e-print http://www.gm.fh-koeln.de/ciopwebpub/Konen22b.d/TR-Rubiks.pdf
+Towards Learning Rubik’s Cube with
+N-tuple-based Reinforcement Learning
+Wolfgang Konen
+Technical Report,
+Computer Science Institute,
+TH Köln,
+University of Applied Sciences,
+Germany
+wolfgang.konen@th-koeln.de
+Sep 2022,
+last update Jan 2023
+Abstract
+This work describes in detail how to learn and solve the Rubik’s cube game (or
+puzzle) in the General Board Game (GBG) learning and playing framework. We cover
+the cube sizes 2x2x2 and 3x3x3. We describe in detail the cube’s state representation,
+how to transform it with twists, whole-cube rotations and color transformations and
+explain the use of symmetries in Rubik’s cube.
+Next, we discuss different n-tuple
+representations for the cube, how we train the agents by reinforcement learning and
+how we improve the trained agents during evaluation by MCTS wrapping.
+We present results for agents that learn Rubik’s cube from scratch, with and without
+MCTS wrapping, with and without symmetries and show that both, MCTS wrapping
+and symmetries, increase computational costs, but lead at the same time to much
+better results. We can solve the 2x2x2 cube completely, and the 3x3x3 cube in the
+majority of the cases for scrambled cubes up to p = 15 (QTM). We cannot yet reliably
+solve 3x3x3 cubes with more than 15 scrambling twists.
+Although our computational costs are higher with MCTS wrapping and with sym-
+metries than without, they are still considerably lower than in the approaches of McAleer
+et al. (2018, 2019) and Agostinelli et al. (2019) who provide the best Rubik’s cube
+learning agents so far.
+1
+arXiv:2301.12167v1  [cs.LG]  28 Jan 2023
+
+Contents
+1
+Introduction
+4
+1.1
+Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.2
+Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Foundations
+6
+2.1
+Conventions and Symbols
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.1.1
+Color arrangement
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.1.2
+Twist and Rotation Symbols . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.1.3
+Twist Types
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.2
+Facts about Cubes
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2.1
+2x2x2 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.2.2
+3x3x3 Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.3
+The Cube State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.4
+Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.4.1
+Twist Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.4.2
+Whole-Cube Rotations (WCR) . . . . . . . . . . . . . . . . . . . . .
+11
+2.4.3
+Color Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+2.5
+Symmetries
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+3
+N-Tuple Systems
+16
+4
+N-Tuple Representions for the Cube
+18
+4.1
+CUBESTATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+4.2
+STICKER
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+4.3
+STICKER2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+4.4
+Adjacency Sets
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+5
+Learning the Cube
+21
+5.1
+McAleer and Agostinelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+5.2
+N-Tuple-based TD Learning
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+5.2.1
+Temporal Coherence Learning (TCL) . . . . . . . . . . . . . . . . . .
+25
+5.2.2
+MCTS
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+5.2.3
+Method Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+6
+Results
+27
+6.1
+Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+6.2
+Cube Solving with MCTS Wrapper, without Symmetries
+. . . . . . . . . . .
+28
+6.3
+Number of Symmetric States . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+6.4
+The Benefit of Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+6.5
+Computational Costs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+7
+Related Work
+32
+2
+
+8
+Summary and Outlook
+33
+A Calculating sloc from fcol
+37
+A.1 2x2x2 cube
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+37
+A.2 3x3x3 cube
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+B N-Tuple Representations for the 3x3x3 Cube
+38
+B.1
+CUBESTATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+B.2
+STICKER
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+39
+B.3
+STICKER2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+39
+B.4
+Adjacency Sets
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+40
+C Hyperparameters
+40
+3
+
+1
+Introduction
+1.1
+Motivation
+Game learning and game playing is an interesting test bed for strategic decision making.
+Games usually have large state spaces, and they often require complex pattern recognition
+and strategic planning capabilities to decide which move is the best in a certain situation.
+If algorithms learn a game (or, even better, a variety of different games) just by self-play,
+given no other knowledge than the game rules, it is likely that they perform also well on
+other problems of strategic decision making.
+In recent years, reinforcement learning (RL) and deep neural networks (DNN) achieved
+superhuman capabilities in a number of competitive games (Mnih et al., 2015; Silver et al.,
+2016). This success has been a product of the combination of reinforcement learning,
+deep learning and Monte Carlo Tree Search (MCTS). However, current deep reinforcement
+learning (DRL) methods struggle in environments with a high number of states and a small
+number of reward states.
+(a)
+(b)
+Figure 1: (a) Scrambled 3x3x3 Rubik’s Cube. (b) 2x2x2 cube in the middle of a twist.
+The Rubik’s cube puzzle is an example of such an environment since the classical
+3x3x3 cube has 4.3 · 1019 states and only one state (the solved cube) has a reward. A
+somewhat simpler puzzle is the 2x2x2 cube with 3.6 · 106 state and again only one reward
+state. Both cubes are shown in Fig. 1.
+The difficult task to learn from scratch how to solve arbitrary scrambled cubes (i.e.
+without being taught by expert knowledge, whether from humans or from computerized
+solvers) was not achievable with DRL methods for a long time. Recently, the works of
+McAleer et al. (2018, 2019) and Agostinelli et al. (2019) provided a breakthrough in that
+direction (see Sec. 5.1 and 7 for details): Their approach DAVI (Deep Approximate Value
+Iteration) learned from scratch to solve arbitrary scrambled 3x3x3 cubes.
+This work investigates whether TD-n-tuple learning with much lower computational de-
+mands can solve (or partially solve) Rubik’s cube as well.
+4
+
+1.2
+Overview
+The General Board Game (GBG) learning and playing framework (Konen, 2019; Konen and
+Bagheri, 2020; Konen, 2022) was developed for education and research in AI. GBG allows
+applying the new algorithm easily to a variety of games. GBG is open source and available
+on GitHub1. The main contribution of this paper is to take the TD-n-tuple approach from
+GBG (Scheiermann and Konen, 2022) that was also successful on other games (Othello,
+ConnectFour) and to investigate this algorithm on various cube puzzles. We will show that
+it can solve the 2x2x2 cube perfectly and the 3x3x3 cube partly. At the same time it has
+drastically reduced computational requirements compared to McAleer et al. (2019). We
+will show that wrapping the base agent with an MCTS wrapper, as it was done by McAleer
+et al. (2019) and Scheiermann and Konen (2022), is essential to reach this success.
+This work is at the same time an in-depth tutorial how to represent a cube and its
+transformations within a computer program such that all types of cube operations can be
+computed efficiently. As another important contribution we will show how symmetries
+(Sec. 2.5, 6.3 and 6.4) applied to cube puzzles can greatly increase sample efficiency and
+performance.
+The rest of this paper is organized as follows: Sec. 2 lays the foundation for Rubik’s
+cube, its state representation, its transformations and its symmetries. In Sec. 3 we in-
+troduce n-tuple systems and how they can be used to derive policies for game-playing
+agents. Sec. 4 defines and discusses several n-tuple representations for the cube. Sec. 5
+presents algorithms for learning the cube: first the DAVI algorithm of McAleer et al. (2019);
+Agostinelli et al. (2019) and then our n-tuple-based TD learning (with extensions TCL and
+MCTS). In Sec. 6 we present the results when applying our n-tuple-based TD learning
+method to the 2x2x2 and the 3x3x3 cube. Sec. 7 discusses related work and Sec. 8 con-
+cludes.
+1https://github.com/WolfgangKonen/GBG
+5
+
+2
+Foundations
+2.1
+Conventions and Symbols
+We consider in this paper two well-known cube types, namely the 2x2x2 cube (pocket
+cube) and the 3x3x3 cube (Rubik’s cube).
+2.1.1
+Color arrangement
+Each cube consists of smaller cubies: 8 corner cubies for the 2x2x2 cube and 8 corner,
+12 edge and 6 center cubies for the 3x3x3 cube. A corner cubie has 3 stickers of different
+color on its 3 faces. An edge cubie has two, a center cubie has one sticker.
+We enumerate the 6 cube faces with
+(ULF) = (Up, Left, Front) and
+(DRB) = (Down, Right, Back).
+We number the 6 colors with 0,1,2,3,4,5. My cube has these six colors
+012 = wbo = (white,blue,orange) in the (ULF)-cubie2 and
+345 = ygr = (yellow,green,red) in the opposing (DRB)-cubie.
+The solved cube in default position has colors (012345) for the faces (ULFDRB), i.e. the
+white color is at the Up face, blue at Left, orange as Front and so on. We can cut the cube
+such that up- and bottom-face can be folded away and have a flattened representation as
+shown in Figure 2.
+w
+b
+o
+g
+r
+y
+Figure 2: The face colors of the default cube in flattened representation
+2.1.2
+Twist and Rotation Symbols
+Twists of cube faces are denoted by uppercase letters U, L, F, D, R, B. Each of these twists
+means a 90◦ counterclockwise rotation.3 If U = U1 is a 90◦ rotation, then U2 is a 180◦
+rotation and U3=U−1 is a 270◦ rotation.
+Whole-cube rotations are denoted by lowercase letters u, ℓ, f. (We do not need d, r, b
+here, because d = u−1, r = ℓ−1 and so on.)
+Further symbols like fc[i], sℓ[i] that characterize a cube state will be explained in Sec. 2.3.
+2.1.3
+Twist Types
+Cube puzzles can have different twist types or twist metrics:
+2We run through the faces of a cubie in counter-clockwise orientation.
+3The rotation is counterclockwise when looking at the respective face
+6
+
+• QTM (quarter turn metric): only quarter twists are allowed: e.g. U1 and U−1.
+• HTM (half turn metric): quarter and half turns (twists) are allowed: e.g. U1, U2, U3.
+By allowed we mean what counts as one move. In QTM we can realize U2 via U U as
+well, but it costs us 2 moves. In HTM, U2 counts as one move.
+The twist type influences God’s number and the branching factor of the game, see
+Sec. 2.2.
+2.2
+Facts about Cubes
+2.2.1
+2x2x2 Cube
+The number of distinct states for the 2x2x2 pocket cube is (Wikipedia, 2022a)
+8! · 37
+24
+= 7! · 36 = 3, 674, 160 ≈ 3.6 · 106
+(1)
+Why this formula? — We have 8 cubies which we can place in 8! ways on the 8 cube
+positions. Each but the last cubie has the freedom to appear in 3 orientations, which gives
+the factor 37 (the last cubie is then in a fixed orientation, the other two orientations would
+yield illegal cube states). – Each of these raw states has the (ygr)-cubie in any of the
+24 possible positions. Or, otherwise speaking, each truly different state appears in 24
+whole-cube rotations. To factor out the whole-cube rotations, we count only the states with
+(ygr)-cubie in its default position (DRB) and divide the number of raw states by 24, q.e.d.
+God’s number: What is the minimal number of moves needed to solve any cube posi-
+tion? – For the 2x2x2 pocket cube, it is 11 in HTM (half-turn metric) and 14 in QTM.
+Branching factor: 3 · 3 = 9 in HTM and 3 · 2 = 6 in QTM.
+2.2.2
+3x3x3 Cube
+The number of distinct states for the 3x3x3 Cube is (Wikipedia, 2022b)
+8! · 37 · 12! · 211
+2
+= 43, 252, 003, 274, 489, 856, 000 ≈ 4.3 · 1019
+(2)
+Why this formula? – We have 8 corner cubies which we can place in 8! ways on the 8
+cube positions. Each but the last cubie has the freedom to appear in 3 orientations, which
+gives the factor 37. We have 12 edge cubies which we can place in 12! ways on the edge
+positions. Each but the last cubie has the freedom to appear in 2 orientations, which gives
+the factor 211. The division by 2 stems from the fact, that neither alone two corner cubies
+may be swapped nor alone two edge cubies may be swapped. Instead, the number of such
+swaps must be even (factor 2).
+God’s Number: What is the minimal number of moves needed to solve any cube
+position? — For the 3x3x3 Rubik’s Cube, it is 20 in HTM (half-turn metric) and 26 in QTM.
+This is a result from Rokicki et al. (2014), see also http://www.cube20.org/qtm/.
+Branching factor: 6 · 3 = 18 in HTM and 6 · 2 = 12 in QTM.
+7
+
+3
+2
+0
+1
+5
+4
+8
+11
+18
+17
+23
+22
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 3: Sticker numbering for the 2x2x2 cube
+2.3
+The Cube State
+A cube should be represented by objects in GBG in such a way that
+(a) cube states that are equivalent are represented by identical objects
+(b) if two cube states are equivalent, it should be easy to check this by comparing their
+objects
+(c) cube transformations are easy to carry out on these objects.
+Condition (a) means that if two twist sequences lead to the same cube state (e.g. U−1
+and UUU), this should result also in identical objects. Condition (b) means, that the equality
+should be easy to check, given the objects. That is, a cube should not be represented by
+its twist sequence.
+A cube state is in GBG represented by abstract class CubeState and has two describ-
+ing members
+fc[i]
+=
+fcol[i]
+(3)
+sℓ[i]
+=
+sloc[i]
+(4)
+fc[i] = fcol[i] denotes the face color at sticker location i. The color is one out of
+0,1,2,3,4,5 for the colors w,b,o,y,g,r.
+sℓ[i] = sloc[i] contains the sticker location of the sticker which is in position i for the
+solved cube d.
+Members fc and sℓ are vectors with 24 (2x2x2 cube) or 48 (3x3x3 cube) elements
+where i denotes the ith sticker location.
+The stickers are numbered in a certain way which is detailed in Figures 3 and 4 for the
+flattened representations of the 2x2x2 and 3x3x3 cube, resp.
+In principle, one of the two members fc and sℓ would be sufficient to characterize a
+state, since the fcol-sloc-relation
+fc[sℓ[i]] = d.fc[i]
+(5)
+holds, where d denotes the default cube. This is because sℓ[i] transports the sticker i
+of the default cube d to location sℓ[i], i.e. it has the color d.fc[i]. That is, we can easily
+8
+
+6
+5
+4
+7
+3
+0
+1
+2
+10
+9
+8
+16
+23
+22
+36
+35
+34
+46
+45
+44
+11
+15
+17
+21
+37
+33
+47
+43
+12
+13
+14
+18
+19
+20
+38
+39
+32
+40
+41
+42
+28
+27
+26
+29
+25
+30
+31
+24
+Figure 4: Sticker numbering for the 3x3x3 cube. We do not number the center cubies, they
+stay invariant under twists.
+Table 1: The three relevant twists for the 2x2x2 cube
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+U twist
+T
+1
+2
+3
+0
+11
+8
+6
+7
+18
+9
+10 17
+12 13 14 15
+16 22 23 19
+20 21
+4
+5
+L twist
+T
+22
+1
+2 21
+5
+6
+7
+4
+3
+0
+10 11
+12 13
+8
+9
+16 17 18 19
+20 14 15 23
+F twist
+T
+7
+4
+2
+3
+14
+5
+6 13
+9
+10 11
+8
+12 18 19 15
+16 17
+0
+1
+20 21 22 23
+U−1
+T −1
+3
+0
+1
+2
+22 23 6
+7
+5
+9
+10
+4
+12 13 14 15
+16 11
+8
+19
+20 21 17 18
+L−1
+T −1
+9
+1
+2
+8
+7
+4
+5
+6
+14 15 10 11
+12 13 21 22
+16 17 18 19
+20
+3
+0
+23
+F−1
+T −1
+18 19 2
+3
+1
+5
+6
+0
+11
+8
+9
+10
+12
+7
+4
+15
+16 17 13 14
+20 21 22 23
+calculate fc given sℓ. With some more effort, it is also possible to calculate sℓ given fc (see
+Appendix A). Although one of these members fc and sℓ would be sufficient, we keep both
+because this allows to better perform assertions or cross checks during transformations.
+Sometime we need the inverse function s−1
+ℓ [i]: Which sticker is at location i? It is easy
+to calculate s−1
+ℓ
+given sℓ with the help of the relation:
+s−1
+ℓ [sℓ[i]] = i
+(6)
+(Note that it is not possible to invert fc, because the face coloring function is not bijective.)
+2.4
+Transformations
+2.4.1
+Twist Transformations
+Each basic twist is a counterclockwise4 rotation of a face by 90◦. Table 1 shows the 2x2x2
+transformation functions for three basic twists. Each twist transformation can be coded in
+two forms:
+1. T[i] (forward transformation): Which is the new location for the sticker being at i
+before the twist?
+4The rotation is counterclockwise when looking at this face.
+9
+
+2
+1
+3
+0
+23
+22
+5
+4
+8
+11
+18
+17
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 5: The default 2x2x2 cube after twist U1
+2. T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands
+in i after the twist?
+Example (read off from column 0 of Table 1): The L-twist transports sticker at 0 to 22:
+T[0] = 22. The (parent) sticker being at location 9 before the L-twist comes to location 0
+after the twist: T −1[0] = 9. Likewise, for the U-twist we have T[0] = 1 and T −1[0] = 3. We
+show in Fig. 5 the default cube after twist U1.
+How can we apply a twist transformation to a cube state programmatically? – We
+denote with f′
+c and s′
+ℓ the new states for fc and sℓ after transformation. The following
+relations allow to calculate the transformed cube state:
+f′
+c[i]
+=
+fc[T −1[i]]
+(7)
+s′
+ℓ[s−1
+ℓ [i]]
+=
+T[i]
+(8)
+Eq. (7) says: The new color for sticker 0 is the color of the sticker which moves into
+location 0 (fc[9] in the case of an L-twist). To explain Eq. (8), we first note that s−1
+ℓ [i] is the
+sticker being at i before the transformation. Then, Eq. (8) says: „The new location for the
+sticker being at i before the transformation is T[i].“ For example, the L-twist transports the
+current sticker at location 0 to the new location T[0] = 22, i. e. s′
+ℓ[0] = 22.
+For the 2x2x2 cube, these 3 twists U, L, F are sufficient, because D=U−1, R=L−1,
+B=F−1. This is because the 2x2x2 cube has no center cubies. For the 3x3x3 cube, we
+need all 6 twists U, L, F, D, R, B because this cube has center cubies.
+In any case, we will show in Sec. 2.4.2 that only one row in Table 1 or Table 2, say T
+for the U-twist, has to be known or established ’by hand’. All other twists and their inverses
+can be calculated programmatically with the help of Eqs. (9)-(15) that will be derived in
+Sec. 2.4.2.
+Table 2: The U twist for the 3x3x3 cube
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+U twist T
+2
+3
+4
+5
+6
+7
+0
+1
+22 23 16 11
+12 13 14 15
+36 17 18 19
+20 21 34 35
+24 25 26 27
+28 29 30 31
+32 33 34 35
+36 37 38 39
+40 41 42 43
+44 45 46 47
+U twist T
+24 25 26 27
+28 29 30 31
+32 33 44 45
+46 37 38 39
+40 41 42 43
+8
+9
+10 47
+10
+
+Table 3: Two basic whole-cube rotations for the 2x2x2 cube
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+u rotation
+T
+1
+2
+3
+0
+11
+8
+9
+10
+18 19 16 17
+15 12 13 14
+21 22 23 20
+6
+7
+4
+5
+f rotation
+T
+7
+4
+5
+6
+14 15 12 13
+9
+10 11
+8
+17 18 19 16
+2
+3
+0
+1
+23 20 21 22
+u−1
+T −1
+3
+0
+1
+2
+22 23 20 21
+5
+6
+7
+4
+13 14 15 12
+10 11
+8
+19
+19 16 17 18
+f−1
+T −1
+18 19 16 17
+1
+2
+3
+0
+11
+8
+9
+10
+6
+7
+4
+5
+15 12 13 14
+21 22 23 20
+Normalizing the 2x2x2 Cube
+As stated above, the 3 twists U, L, F are sufficient
+for the 2x2x2 cube. Therefore, the (DRB)-cubie will never leave its place, whatever the
+twist sequence formed by U, L, F is. The (DRB)-cubie has the stickers (12, 16, 20), and we
+can check in Table 1 that columns (12, 16, 20) are always invariant. If we have an arbitrary
+initial 2x2x2 cube state, we can normalize it by applying a whole-cube rotation such that
+the (ygr)-cubie moves to the (DRB)-location.
+Normalizing the 3x3x3 Cube
+In the case of the 3x3x3 cube, all center cubies
+will be not affected by any twist sequence. Therefore, we normalize a 3x3x3 cube state
+by applying initially a whole-cube rotation such that the center cubies are in their normal
+position (i.e. white up, blue left and so on).
+2.4.2
+Whole-Cube Rotations (WCR)
+Each basic whole-cube rotation (WCR) is a counterclockwise rotation of the whole cube
+around the u, l, f-axis by 90◦. Table 3 shows two of the 2x2x2 transformation functions for
+basic whole-cube rotations. Each rotation can be coded in two forms:
+1. T[i] (forward transformation): Which is the new location for the sticker being at i
+before the twist?
+2. T −1[i] (inverse transformation): Which is the (parent) location of the sticker that lands
+in i after the twist?
+Besides the basic rotation u there is also u2 (180◦) and u3 = u−1 (270◦ = −90◦).
+All whole-cube rotations can be generated from these two forward rotations u and f:
+First, we calculate the inverse transformations via
+T −1[T[i]] = i
+(9)
+where T is a placeholder for u or f. Next, we calculate the missing base rotation ℓ (counter-
+clockwise around the left face) as
+ℓ = fuf−1
+(10)
+We use here the programm-code-oriented notation „first trafo first“: Eq. (10) reads as
+„first f, then u, then f−1“.5
+5In programm code the relation would read cs.fTr(1).uTr().fTr(3). This is „first trafo first“, because
+each transformation is applied to the cube state object to the left and returns the transformed cube state object.
+11
+
+Table 4: All 24 whole-cube rotations (in first-trafo-first notation)
+number
+first rotation
+∗ u0
+∗ u1
+∗ u2
+∗ u3
+00-03
+id (white up)
+id
+u
+u2
+u3
+04-07
+f (green up)
+f
+fu
+fu2
+fu3
+08-11
+f2 (yellow up)
+f2
+f2u
+f2u2
+f2u3
+12-15
+f−1 (blue up)
+f−1
+f−1u
+f−1u2
+f−1u3
+16-19
+ℓ (orange up)
+ℓ
+ℓu
+ℓu2
+ℓu3
+20-23
+ℓ−1 (red up)
+ℓ−1
+ℓ−1u
+ℓ−1u2
+ℓ−1u3
+Table 5: The 24 inverse whole-cube rotations (in first-trafo-first notation)
+number
+first rotation
+∗ u0
+∗ u1
+∗ u2
+∗ u3
+00-03
+id (white up)
+id
+u3
+u2
+u1
+04-07
+f (green up)
+f−1
+ℓu3
+fu2
+ℓ−1u
+08-11
+f2 (yellow up)
+f2
+f2u
+f2u2
+f2u3
+12-15
+f−1 (blue up)
+f
+ℓ−1u3
+f−1u2
+ℓu
+16-19
+ℓ (orange up)
+ℓ−1
+f−1u3
+ℓu2
+fu
+20-23
+ℓ−1 (red up)
+ℓ
+fu−1
+ℓ−1u2
+f−1u
+The other basic whole-cube rotations d, r, b are not needed, because d = u−1, r = ℓ−1
+and b = f−1.
+The basic whole-cube rotations are rotations of the whole cube around just one axis.
+But there are also composite whole-cube rotations which consists of a sequence of basic
+rotations.
+How many different (composite) rotations are there for the cube? – A little thought
+reveals that there are 24 of them: To be specific, we consider the default cube where we
+have 4 rotations with the white face up, 4 with the blue face up, and so on. In total we have
+6 · 4 = 24 rotations since there are 6 faces. Table 4 lists all of them, togehter with the WCR
+numbering convention used in GBG.
+Sometimes we need the inverse whole-cube rotations which are given in Table 5. In
+this table, we read for example from the element with number 5, that the WCR with key 5
+(which is fu according to Table 4) has the inverse WCR ℓu3 such that
+fu ℓu3 = id
+holds.
+For convenience, we list in Table 6 the <Key, InverseKey> relation. For example, the
+trafo with Key=5 (fu) has the inverse trafo with InverseKey=19 (ℓu3). Note that there are
+10 whole-cube rotations which are their own inverse.
+Generating all twists from U twist
+With the help of WCRs we can generate the other
+twists from the U twist only: We simply rotate the face that we want to twist to the up-face,
+12
+
+Table 6: Whole-cube rotations: <Key, InverseKey> relation
+key
+0 1 2 3
+4
+5
+6
+7
+8 9 10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+inv key
+0 3 2 1
+12 19 6 21
+8 9 10 11
+4
+23 14 17
+20 15 18 05
+16
+7
+22 13
+apply the U twist and rotate back. This reads in first-trafo-first notation:
+L = f−1Uf
+(11)
+F = ℓUℓ−1
+(12)
+D = f2Uf2
+(13)
+R = fUf−1
+(14)
+B = ℓ−1Uℓ
+(15)
+Thus, given the U twist from Table 1 or Table 2 and the basic WCRs given in Table 3 and
+Eq. (10), we can calculate all other forward transformations with the help of Eqs. (11)–(15).
+Then, all inverse transformations are calculable with the help of Eq. (9).
+2.4.3
+Color Transformations
+Color transformations are special transformations that allow to discover non-trivial symmet-
+ric (equivalent) states.
+One way to describe a color transformation is to select a valid color permutation and to
+paint each sticker with the new color according to this color permutation. This is of course
+nothing one can do with a real cube without destroying or altering it, but it is a theoretical
+concept leading to an equivalent state.
+Another way of looking at it is to record the twist sequence that leads from the default
+cube to a certain scrambled cube state. Then we go back to the default cube, make at
+first a whole-cube rotation (leading to a color-transformed default cube) and then apply the
+recorded twist sequence to the color-transformed default cube.
+In any case, the transformed cube will be usually not in its normal position, so we apply
+finally a normalizing operation to it.
+What are valid color permutations? – These are permutations of the cube colors reach-
+able when applying one of the available 24 WCRs (Table 4) to the default cube. For exam-
+ple, if we apply WCR f (number 04) to the default cube, we get
+g
+w
+o
+y
+r
+b
+Figure 6: The color transformation according to WCR f (number 04)
+that is, g (green) is the new color for each up-sticker that was w (white) before and so
+on. The colors o and r remain untouched under this color permutation. [However, other
+transformations like fu, fu2 and fu3 will change every color.]
+13
+
+2
+1
+3
+0
+23
+22
+5
+4
+8
+11
+18
+17
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 7: The cube of Fig. 5 before color transformation.
+16
+19
+17
+18
+20
+23
+2
+1
+11
+10
+13
+12
+3
+0
+8
+9
+14
+15
+21
+22
+4
+7
+5
+6
+8
+2
+9
+1
+4
+7
+14
+11
+18
+17
+23
+0
+5
+6
+15
+10
+19
+16
+20
+3
+21
+13
+22
+12
+(a)
+(b)
+Figure 8: The cube of Fig. 7 with color transformation from Fig 6: (a) before normalization,
+(b) after normalization.
+How can we apply a color transformation to a cube state programmatically? – We
+denote with f′ and s′
+ℓ the new states for f and sℓ after transformation.
+The following
+relations allow to calculate the transformed cube state:
+f′
+c[i]
+=
+c[fc[i]]
+(16)
+s′
+ℓ[s−1
+ℓ [i]]
+=
+T[i]
+(17)
+where c[] is the 6-element color trafo vector (holding the new colors for current colors 0:w,
+1:b, ..., 5:r) and T is the 24- or 48-element vector of the WCR that produces this color
+transformation. Eq. (16) is simple: If a certain sticker has color 0 (w, white) before the color
+transformation, then it will get the new color c[0], e.g. 4 (g, green), after the transformation.
+Eq. (17) looks complicated, but it has a similar meaning as in the twist trafo: Take i = 0 as
+example: The new place for the sticker being at 0 before the trafo (and coming from s−1
+ℓ [0])
+is T[0]. Therefore, we write the number T[0] into s′
+ℓ[s−1
+ℓ [0]].
+A color transformation example is shown in Figs. 7 and 8. Fig. 7 is just a replication
+of Fig. 5 showing a default cube after U1 twist. The color transformation number 04 applied
+to the cube of Fig. 7 is shown in Fig. 8 (a)-(b) in two steps:
+(a) The stickers are re-painted and re-numbered (white becomes green, blue becomes
+white and so on). The structure of coloring is the same as in Fig. 7. Now the (DRB)-
+cubie is no longer the (ygr)-cubie, it does not carry the numbers (12,16,20).
+14
+
+(b) We apply the proper WCR that brings the (ygr)-cubie back to the (DRB)-location.
+Compared to (a), each 4-sticker cube face is just rotated to another face, but not
+changed internally. We can check that the (DRB)-location now carries again the num-
+bers (12,16,20), as in Fig. 7 and as it should for a normalized cube.
+2.5
+Symmetries
+Symmetries are transformations of the game state (and the attached action, if applicable)
+that lead to equivalent states.
+That is, if s is a certain state with value V (s), then all
+states ssym being symmetric to s have the same value V (ssym) = V (s) because they are
+equivalent. Equivalent means: If s can be solved by a twist sequence of length n, then
+ssym can be solved by an equivalent twist sequence of same length n.
+In the case of Rubik’s cube, all whole-cube rotations (WCRs) are symmetries because
+they do not change the value of a state. But whole-cube rotations are ’trivial’ symmetries
+because they are usually factored out by the normalization of the cube: After 2x2x2 cube
+normalization, which brings the (ygr)-cubie in a certain position, or after 3x3x3 cube nor-
+malization, which brings the center cubies in certain faces, all WCR-symmetric states are
+transformed to the same state.
+Non-trivial symmetries are all color transformations (Sec. 2.4.3): In general, color trans-
+formations transform a state s to a truly different state ssym, even after cube normalization.6
+Since there are 24 color transformations in Rubik’s cube, there are also 24 non-trivial sym-
+metries (including self).
+Symmetries are useful to learn to solve Rubik’s cube for two reasons: (a) to accelerate
+learning and (b) to smooth an otherwise noisy value function.
+(a) Accelerated learning: If a state s (or state-action pair) is observed, not only the
+weights activated by that state are updated, but also the weights of all symmetric states
+ssym, because they have the same V (ssym) = V (s) and thus the same reward. In this
+way, a single observed sample is connected with more weight updates (better sample
+efficiency).
+(b) Smoothed value function: By this we mean that the value function V (s) is replaced
+by
+V (sym)(s) =
+1
+|Fs|
+�
+s′∈Fs
+V (s′)
+(18)
+where Fs is the set of states being symmetric to s. If V (s) were the ideal value function,
+both terms V (s) and V (sym)(s) would be the same.7 But in a real n-tuple network, V (s)
+is non-ideal due to n-tuple-noise (cross-talk from other states that activate the same
+n-tuple LUT entries). If we average over the symmetric states s′ ∈ Fs, the noise will be
+dampened.
+6In rare cases – e.g. for the solved cube – the transformed state may be identical to s or to another
+symmetry state, but this happens seldom for sufficiently scrambled cubes, see Sec. 6.3.
+7because all V (s′) in Eq. (18) are the same for an ideal V
+15
+
+The downside of symmetries is their computational cost: In the case of Rubik’s cube,
+the calculation of color transformations is a costly operation. On the other hand, the number
+of necessary training episodes to reach a certain performance may be reduced. In the
+end, the use of symmetries may pay off, because the total training time may be reduced as
+well. In any case, we will have a better sample efficiency, since we learn more from each
+observed state or state-action pair. Secondly, the smoothing effect introduced with Eq. (18)
+can lead to better overall performance, because the smoothed value function provides a
+better guidance on the path towards the solved cube.
+In order to balance computation time, GBG offers the option to select with nSym the
+number of symmetries actually used. If we specify for example nSym = 8 in GBG’s Rubik’s
+cube implementation, then the state itself and 8 – 1 = 7 random other (non-id) color trans-
+formations will be selected. The resulting set Fs of 8 states is then used for weight update
+and value function computation.
+3
+N-Tuple Systems
+N-tuple systems coupled with TD were first applied to game learning by Lucas (2008), al-
+though n-tuples were already introduced by Bledsoe and Browning (1959) for character
+recognition purposes. The remarkable success of n-tuples in learning to play Othello (Lu-
+cas, 2008) motivated other authors to benefit from this approach for a number of other
+games.
+The main goal of n-tuple systems is to map a highly non-linear function in a low di-
+mensional space to a high dimensional space where it is easier to separate ‘good’ and
+‘bad’ regions. This can be compared to the kernel trick of support-vector machines. An
+n-tuple is defined as a sequence of n cells of the board. Each cell can have m positional
+values representing the possible states of that cell.8 Therefore, every n-tuple will have a
+(possibly large) look-up table indexed in form of an n-digit number in base m. Each entry
+corresponds to a feature and carries a trainable weight. An n-tuple system is a system
+consisting of k n-tuples. As an example we show in Fig. 9 an n-tuple system consisting of
+four 8-tuples.
+Let Θ be the vector of all weights θi of the n-tuple system.9 The length of this vector
+may be large number, e.g. mnk, if all k n-tuples have the same length n and each cell
+has m positional values. Let Φ(s) be a binary vector of the same length representing the
+feature occurences in state s (that is, Φi(s) = 1 if in state s the cell of a specific n-tuple as
+indexed by i has the positional value as indexed by i, Φi(s) = 0 else). The value function
+of the n-tuple network given state s is
+V (s) = σ (Φ(s) · Θ)
+(19)
+with transfer function σ which may be a sigmoidal function or simply the identity function.
+8A typical example is a 2-player board game, where we usually have 3 positional values {0: empty, 1:
+player1, 2: player2 }. But other, user-defined values are possible as well.
+9The index i indexes three qualities: an n-tuple, a cell in this n-tuple and a positional value for this cell.
+16
+
+Figure 9: Example n-tuples: We show 4 random-walk 8-tuples on a 6x7 board. The tuples are
+selected manually to show that not only snake-like shapes are possible, but also bifurcations
+or cross shapes. Tuples may or may not be symmetric.
+An agent using this n-tuple system derives a policy from the value function in Eq. (19)
+as follows: Given state s and the set A(s) of available actions in state s, it applies with a
+forward model f every action a ∈ A(s) to state s, yielding the next state s′ = f(s, a). Then
+it selects the action that maximizes V (s′).
+Each time a new agent is constructed, all n-tuples are either created in fixed, user-
+defined positions and shapes, or they are formed by random walk. In a random walk, all
+cells are placed randomly with the constraint that each cell must be adjacent10 to at least
+one other cell in the n-tuple.
+Agent training proceeds in the TD-n-tuple algorithm as follows: Let s′ be the actual
+state generated by the agent and let s be the previous state generated by this agent. TD(0)
+learning adapts the value function with model parameters Θ through (Sutton and Barto,
+1998)
+Θ ← Θ + αδ∇ΘV (s)
+(20)
+Here, �� is the learning rate and V is in our case the n-tuple value function of Eq. (19). δ is
+the usual TD error (Sutton and Barto, 1998) after the agent has acted and generated s′:
+δ = r + γV (s′) − V (s)
+(21)
+where the sum of the first two terms, reward r plus the discounted value γV (s′), is the
+desirable target for V (s).
+10The form of adjacency, e. g. 4- or 8-point neighborhood or any other (might be cell-dependent) form of
+adjacency, is user-defined.
+17
+
+0
+3
+5
+6
+6
+5
+6
+3
+4
+4
+5
+6
+54
+N-Tuple Representions for the Cube
+In order to apply n-tuples to cubes, we have to define a board in one way or the other on
+which we can place the n-tuples. This is not as straightforward as in other board games, but
+we are free to invent abstract boards. Once we have defined a board, we can number the
+board cells k = 0, . . . , K−1 and translate a cube state into a BoardVector: A BoardVector
+b is a vector of K non-negative integer numbers bk ∈ {0, . . . , Nk − 1}. Each k represents
+a board cell and every board cell k has a predefined number Nk of position values.11
+A BoardVector is useful to calculate the feature occurence vector Φ(s) in Eq. (19) for
+a given n-tuple set: If an n-tuple contains board cell k, then look into bk to get the position
+value for this cell k. Set Φi(s) = 1 for that index i that indexes this n-tuple cell and this
+position value.
+In the following we present different options for boards and BoardVectors. We do this
+mainly for the 2x2x2 cube, because it is somewhat simpler to explain. But the same ideas
+apply to the 3x3x3 cube as well, they are just a little bit longer. Therefore, we defer the
+lengthy details of the 3x3x3 cube to Appendix B.
+4.1
+CUBESTATE
+A natural way to translate the cube state into a board is to use the flattened representation
+of Fig. 11 as the board and extract from it the 24-element vector b, according to the given
+numbering. The kth element bk represents a certain cubie face location and gets a number
+from {0, . . . , 5} according to its current face color fc. The solved cube is for example
+represented by b = [0000 1111 2222 . . . 5555].
+This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG-
+implementation means: Each board vector is a copy of fcol, the face colors of all cubie
+faces. fcol is also the vector that uniquely defines each cube state. An upper bound of
+possible combinations for b is 624 = 4.7 · 1018. If we factor out the (DRB)-cubie, which
+always stays at its home position, we can reduce this to 21 board cells with 6 positional
+values, leading to 621 = 2.1 · 1016 weights. Both numbers are of course way larger than
+the true number of distinct states (Sec. 2.2.1) which is 3.6 · 106. This is because most of
+the combinations are dead weights in the n-tuple LUTs, they will never be activated during
+game play.
+The dead weights occur because many combinations are not realizable, e.g. three
+white faces in one cubie or any of the 63 − 8 · 3 = 192 cubie-face-color combinations that
+are not present in the real cube. The problem is that the dead weights are scattered in a
+complicated way among the active weights and it is thus not easy to factor them out.
+4.2
+STICKER
+McAleer et al. (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie
+faces) are enough. To characterize the full 3x3x3 cube, we need only one (not 2 or 3) sticker
+11In GBG package ntuple2 (base for agent TDNTuple3Agt), all Nk have to be the same.
+In package
+ntuple4( base for agent TDNTuple4Agt), numbers Nk may be different for different k.
+18
+
+(a) Top view
+(b) Bottom view
+Figure 10: The sticker representation used to reduce dimensionality: Stickers that are used
+are shown in white, whereas ignored stickers are dark blue (from McAleer et al. (2019)).
+3
+2
+0
+1
+5
+4
+8
+11
+18
+17
+23
+22
+6
+7
+9
+10
+19
+16
+20
+21
+14
+13
+15
+12
+Figure 11: Tracked stickers for the 2x2x2 cube (white), while ignored stickers are blue.
+for every of the 20 cubies, as shown in Fig. 10. This is because the location of one sticker
+uniquely defines the location and orientation of that cubie. We name this representation
+STICKER in GBG.
+Translated to the 2x2x2 cube, this means that 8 stickers are enough because we have
+only 8 cubies. We may for example track the 4 top stickers 0,1,2,3 plus the 4 bottom
+stickers 12,13,14,15 as shown in Fig. 11 and ignore the 16 other stickers. Since we always
+normalize the cube such that the (DRB)-cubie with sticker 12 stays in place, we can reduce
+this even more to 7 stickers (all but sticker 12).
+How to lay out this representation as a board? – McAleer et al. (2019) create a rect-
+angular one-hot-encoding board with 7 × 21 = 147 cells (7 rows for the stickers and 21
+columns for the locations) carrying only 0’s and 1’s. This is fine for the approach of McAleer
+et al. (2019), where they use this board as input for a DNN, but not so nice for n-tuples.
+Without constraints, such a board amounts to 2147 = 1.7 · 1044 combinations, which is
+unpleasantly large (much larger than in CUBESTATE).12
+STICKER has more dead weights than CUBESTATE, so it seems like a step back. But
+the point is, that the dead weights are better structured: If for example sticker 0 appears at
+column 1 then this column and the two other columns for the same cubie are automatically
+12A possible STICKER BoardVector for the default cube would read b = [1000000 0100000 0010000 . . . ],
+meaning that location 0 has the first sticker, location 1 has the second sticker, and so on. In any STICKER
+BoardVector there are only 7 columns carrying exactly one 1, the other carry only 0’s. Every row carries exactly
+one 1.
+19
+
+Table 7: The correspondence corner location ↔ STICKER2 for the solved cube. The yellow
+colored cells show the location of the 7 (2x2x2) and 8 (3x3x3) corner stickers that we track.
+2x2x2
+location
+0 1 2 3
+4
+5
+6
+7
+8
+9
+10 11
+12 13 14 15
+16 17 18 19
+20 21 22 23
+3x3x3
+location
+0 2 4 6
+8 10 12 14
+16 18 20 22
+24 26 28 30
+32 34 36 38
+40 42 44 46
+STICKER2
+corner
+a b c d
+a
+d
+h
+g
+a
+g
+f
+b
+e
+f
+g
+h
+e
+c
+b
+f
+e
+h
+d
+c
+face ID
+1 1 1 1
+2
+3
+2
+3
+3
+2
+3
+2
+1
+1
+1
+1
+2
+2
+3
+2
+3
+3
+2
+3
+forbidden for all other stickers. Likewise, if sticker 1 is placed in another column, another set
+of 3 columns is forbidden, and so on. We can use this fact to form a much more compact
+representation STICKER2.
+4.3
+STICKER2
+As the analysis in the preceding section has shown, the 21 location columns of STICKER
+cannot carry the tracked stickers in arbitrary combinations. Each cubie (represented by 3
+columns in STICKER) carries only exactly one sticker. We can make this fact explicit by
+choosing another representation for the 21 locations:
+corner location = (corner cubie, face ID).
+That is, each location is represented by a pair: corner cubie a,b,c,d,f,g,h (we number the
+top cubies with letters a,b,c,d and the bottom cubies with letters e,f,g,h and omit e because
+it corresponds to the (DRB)-cubie) and a face ID. To number the faces with a face ID, we
+follow the convention that we start at the top (bottom) face with face ID 1 and then move
+counter-clockwise around the corner cubie to visit the other faces (2,3). Table 7 shows the
+explicit numbering in this new representation.
+To represent a state as board vector we use now a much smaller board shown in
+Table 8: Each cell in the first row has 7 position values (the letters) and each cell in the
+second row has 3 position values (the face IDs). We show in Table 8 the board vector for
+the default cube, b = [abcdfgh 1111111]. Representation STICKER2 allows for 77 · 37 =
+1.8 · 109 combinations in total, which is much smaller than STICKER and CUBESTATE.
+Table 8: STICKER2 board representation for the default 2x2x2 cube. For the BoardVector,
+cells are numbered row-by-row from 0 to 16.
+corner
+a
+b
+c
+d
+f
+g
+h
+7 positions
+face ID
+1
+1
+1
+1
+1
+1
+1
+3 positions
+STICKER2 has some dead weights remaining, because the combinations can carry
+the same letter multiple times, which is not allowed for a real cube state. But this rate of
+dead weights is tolerable.
+It turns out that STICKER2 is in all aspects better than CUBESTATE or STICKER.
+Therefore, we will only report the results for STICKER2 in the following.
+20
+
+4.4
+Adjacency Sets
+To create n-tuples by random walk, we need adjacency sets (sets of neighbors) to be
+defined for every board cell k.
+For CUBESTATE, the board is the flattened representation of the 2x2x2 cube (Fig. 3).
+The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh-
+bors if they share a common edge on the cube, i.e. are neighbors on the cube.
+For STICKER2, the board consists of 16 cells shown in Table 8. Here, the adjacency
+set for cell k contains all other cells different from k.
+Again, the details of ideas similar to Sec. 4.1–4.4, but now for the 3x3x3 cube, are
+shown in Appendix B.1–B.4.
+5
+Learning the Cube
+5.1
+McAleer and Agostinelli
+The works of McAleer et al. (2018, 2019) and Agostinelli et al. (2019) contain up to now
+the most advanced methods for learning to solve the cube from scratch. Agostinelli et al.
+(2019) introduces the cost-to-go function for a general Marko decision process
+J(s) = min
+a∈A(s)
+�
+s′
+P a(s, s′)
+�
+ga(s, s′) + γJ(s′)
+�
+(22)
+where P a(s, s′) is the probability of transitioning from state s to s′ by taking action a and
+ga(s, s′) is the cost for this transition. In the Rubik’s cube case, we have deterministic tran-
+sitions, that is s′ = f(s, a) is deterministically prescribed by a forward model f. Therefore,
+the sum reduces to one term and we specialize to γ = 1. Furthermore, we set ga(s, s′) = 1,
+because only the length of the solution path counts, so that we get the simpler equation
+J(s) = min
+a∈A(s)
+�
+1 + J(s′)
+�
+with
+s′ = f(s, a).
+(23)
+Here, A(s) is the set of available actions in state s. We additionally set J(s∗) = 0 if s∗
+is the solved cube. To better understand Eq. (23) we look at a few examples: If s1 is a state
+one twist away from s∗, Eq. (23) will find this twist and set J(s1) = 1. If s2 is a state two
+twists away from s∗ and all one-twist states have already their correct labels J(s1) = 1,
+then Eq. (23) will find the twist leading to a s1 state and set J(s2) = 1 + 1 = 2. While
+iterations proceed, more and more states (being further away from s∗) will be correctly
+labeled, once their preceding states are correctly labeled. In the end we should ideally
+have
+J(sn) = n.
+However, the number of states for Rubik’s cube is too large to store them all in tabular
+form. Therefore, McAleer et al. (2019) and Agostinelli et al. (2019) approximate J(s) with
+a deep neural network (DNN). To train such a network in the Rubik’s cube case, they
+21
+
+Algorithm 1 DAVI algorithm (from Agostinelli et al. (2019)). Input: B: batch size, K:
+maximum number of twists, M: training iterations, C: how often to check for convergence,
+ϵ: error threshold. Output: Θ, the trained neural network parameters.
+1: function DAVI(B, K, M, C, ϵ)
+2:
+Θ ← INITIALIZENETWORKPARAMETERS
+3:
+ΘC ← Θ
+4:
+for m = 1, . . . , M do
+5:
+X ←GENERATESCRAMBLEDSTATES(B, K)
+▷ B scrambled cubes
+6:
+for xi ∈ X do
+7:
+yi ← mina∈A(s) [1 + jΘC(f(xi, a))]
+▷ cost-to-go function, Eq. (23)
+8:
+(Θ, loss) ← TRAIN(jΘ, X, y)
+▷ loss = MSE(jΘ(xi), yi)
+9:
+if (m mod C = 0 & loss < ϵ) then
+10:
+ΘC ← Θ
+11:
+return Θ
+introduce Deep Approximate Value Iteration (DAVI)13 shown in Algorithm 1. The network
+output jΘ(s) is trained in line 8 to approximate the (unknown) cost-to-go J(s) for every
+state s = xi. The main trick of DAVI is, as Agostinelli et al. (2019) write: „For learning to
+occur, we must train on a state distribution that allows information to propagate from the
+goal state to all the other states seen during training. Our approach for achieving this is
+simple: each training state xi is obtained by randomly scrambling the goal state ki times,
+where ki is uniformly distributed between 1 and K. During training, the cost-to-go function
+first improves for states that are only one move away from the goal state. The cost-to-go
+function then improves for states further away as the reward signal is propagated from the
+goal state to other states through the cost-to-go function.“
+Agostinelli et al. (2019) use in Algorithm 1 two sets of parameters to train the DNN: the
+parameters Θ being trained and the parameters ΘC used to obtain improved estimates
+of the cost-to-go function. If they did not use this two separate sets, performance often
+„saturated after a certain point and sometimes became unstable. Updating ΘC only after
+the error falls below a threshold ϵ yields better, more stable, performance.“ (Agostinelli
+et al., 2019) To train the DNN, they used M = 1 000 000 iterations, each with batch size
+B = 10 000. Thus, the trained DNN has seen ten billion cubes (1010) during training, which
+is still only a small subset of the 4.3 · 1019 possible cube states.
+The heuristic function of the trained DNN alone cannot solve 100% of the cube states.
+Especially for higher twist numbers ki, an additional solver or search algorithm is needed.
+This is in the case of McAleer et al. (2019) a Monte Carlo Tree Search (MCTS), similar to
+AlphaZero (Silver et al., 2017), which uses the DNN as the source for prior probabilities.
+Agostinelli et al. (2019) use instead a variant of A∗-search, which is found to produce
+solutions with a shorter path in a shorter runtime than MCTS.
+13More precisely, McAleer et al. (2019) use Autodidactic Iteration (ADI), a precursor to DAVI, very similar to
+DAVI, just a bit more complicated to explain. Therefore, we describe here only DAVI.
+22
+
+Algorithm 2 TD-n-tuple algorithm for Rubik’s cube. Input: pmax: maximum number of
+twists, M: training iterations, Etrain: maximum episode length during training, c: nega-
+tive cost-to-go, Rpos: positive reward for reaching the solved cube s∗, α: learning rate.
+jΘ(s): n-tuple network value prediction for state s. Output: Θ, the trained n-tuple network
+parameters.
+1: function TDNTUPLE(pmax, M, Etrain, c, Rpos)
+2:
+Θ ← INITIALIZENETWORKPARAMETERS
+3:
+for m = 1, . . . , M do
+4:
+p ∼ U(1, . . . , pmax)
+▷ Draw p uniformly random from {1, 2, . . . , pmax}
+5:
+s ← SCRAMBLESOLVEDCUBE(p)
+▷ start state
+6:
+for k = 1, . . . , Etrain do
+7:
+snew ← arg max
+a∈A(s)
+V (s′)
+with
+s′ = f(s, a)
+and
+8:
+V (s′) = c +
+� Rpos
+if
+s′ = s∗
+jΘ(s′)
+if
+s′ ̸= s∗
+9:
+Train network jΘ with Eq. (20) to bring V (s) closer to target T = V (snew):
+V (s) ← V (s) + α(T − V (s))
+10:
+s ← snew
+11:
+if (s = s∗) then
+12:
+break
+▷ break out of k-loop
+13:
+return Θ
+23
+
+5.2
+N-Tuple-based TD Learning
+To solve the Rubik’s cube in GBG we use an algorithm that is on the one hand inspired by
+DAVI, but on the other hand more similar to traditional reinforcement learning schemes like
+temporal difference (TD) learning. In fact, we want to use in the end the same TD-FARL
+algorithm (Konen and Bagheri, 2021) that we use for all other GBG games.
+We show in Algorithm 2 our method, that we will explain in the following, highlighting
+also the similarities and dissimilarities to DAVI.
+First of all, instead of minimizing the positive cost-to-go as in DAVI, we maximize in
+lines 7-8 a value function V (s′) with a negative cost-to-go. This maximization is functionally
+equivalent, but more similar to the usual TD-learning scheme. The negative cost-to-go, e.g.
+c = −0.1, plays the role of the positive 1 in Eq. (23).
+Secondly, we replace the DNN of DAVI by the simpler-to-train n-tuple network jΘ with
+STICKER2 representation as described in Sec. 3 and 4.
+That is, each time jΘ(s′) is
+requested, we first calculate for state s′ the BoardVector in STICKER2 representation,
+then the occurence vector Φ(s′) and the value function V (s′) according to Eq. (19).
+The central equations for V (s′) in Algorithm 2, lines 7-8, work similar to Eq. (23) in
+DAVI: If s = s1 is a state one twist away from s∗, the local search in arg max V (s′) will find
+this twist and the training step in line 9 moves V (s) closer to c+Rpos.14 Likewise, neighbors
+s2 of s1 will find s1 and thus move V (s2) closer to 2c + Rpos. Similar for s3, s4, . . . under
+the assumption that a ’known’ state is in the neighborhood. We have a clear gradient on
+the path towards the solved cube s∗. If there are no ’known’ states in the neighborhood
+of sn, we get for V (sn) what the net maximally estimates for all those neighbors. We pick
+the neighbor with the highest estimate, wander around randomly until we hit a state with a
+’known’ neighbor or until we reach the limit Etrain of too many steps.
+Note that Algorithm 2 is different from DAVI insofar that it follows the path s → s′ → . . .
+as prescribed by the current V , which may lead to a state sequence ’wandering in the
+unknown’ until Etrain is reached. In contrast to that, DAVI generates many start states s0
+drawn from the distribution of training set states and trains the network just on pairs (s0, T),
+i.e. they do just one step on the path. We instead follow the full path, because we want
+the training method for Rubik’s cube to be as similar as possible to the training method for
+other GBG games.15
+Algorithm 2 is basically the same algorithm as GBG uses for other games. The only
+differences are (i) the cube-specific start state selection borrowed from DAVI (a 1-twist start
+state has the same probability as a 10-twist start state) and (ii) the cube-specific reward in
+line 8 of Algorithm 2 with its negative cost-go-go c which is however a common element of
+many RL rewards.
+Algorithm 2 currently learns with only one parameter vector Θ. However, it could be
+extended as in DAVI to two parameter vectors Θ and ΘC. The weight training step in line
+14It is relevant, that Rpos is a positive number, e.g. 1.0 (and not 0, as it was for DAVI). This is because we
+start with an initial n-tuple network with all weights set to 0, so the initial response of the network to any state
+is 0.0. Thus, if Rpos were 0, a one-twist state would see all its neighbors (including s∗) initially as responding
+0.0 and would not learn the right transition to s∗. With Rpos = 1.0 it will quickly find s∗.
+15We note in passing that we tested the DAVI variant with Etrain = 1 for our TD-n-tuple method as well.
+However, we found that this method gave much worse results, so we stick with our GBG method here.
+24
+
+9 is done with the help of Eq. (20) for Θ using the error signal δ of Eq. (21).
+There are two extra elements, TCL and MCTS, that complete our n-tuple-based TD
+learning. They are described in the next two subsections.
+5.2.1
+Temporal Coherence Learning (TCL)
+The TCL algorithm developed by Beal and Smith Beal and Smith (1999) is an extension
+of TD learning. It replaces the global learning rate α with the weight-individual product
+ααi for every weight θi. Here, the adjustable learning rate αi is a free parameter set by a
+pretty simple procedure: For each weight θi, two counters Ni and Ai accumulate the sum
+of weight changes and the sum of absolute weight changes. If all weight changes have the
+same sign, then αi = |Ni|/Ai = 1, and the learning rate stays at its upper bound. If weight
+changes have alternating signs, then the global learning rate is probably too large. In this
+case, αi = |Ni|/Ai → 0 for t → ∞, and the effective learning rate will be largely reduced
+for this weight.
+In our previous work (Bagheri et al., 2015) we extended TCL to αi = g(|Ni|/Ai) where
+g is a transfer function being either the identity function (standard TCL) or an exponential
+function g(x) = eβ(x−1). It was shown in Bagheri et al. (2015) that TCL with this exponential
+transfer function leads to faster learning and higher win rates for the game ConnectFour.
+5.2.2
+MCTS
+We use Monte Carlo Tree Search (MCTS) (Browne et al., 2012) to augment our trained
+network during testing and evaluation. This is the method also used by McAleer et al.
+(2019) and by AlphaGo Zero (Silver et al., 2017), but they use it also during training.
+MCTS builds iteratively a search tree starting with a tree containing only the start state
+s0 as the root node. Until the iteration budget is exhausted, MCTS does the following: In
+every iteration we start from the root node and select actions following the tree policy until
+we reach a yet unexpanded leaf node sℓ. The tree policy is implemented in our MCTS
+wrapper according to the UCB formula (Silver et al., 2017):
+anew
+=
+arg max
+a∈A(s)
+�W(s, a)
+N(s, a) + U(s, a)
+�
+(24)
+U(s, a)
+=
+cpuctP(s, a)
+�
+ε + �
+b∈A(s) N(s, b)
+1 + N(s, a)
+(25)
+Here, W(s, a) is the accumulator for all backpropagated values that arrive along branch
+a of the node that carries state s. Likewise, N(s, a) is the visit counter and P(s, a) the prior
+probability. A(s) is the set of actions available in state s. ε is a small positive constant for
+the special case �
+b N(s, b) = 0: It guarantees that in this special case the maximum of
+U(s, a) is given by the maximum of P(s, a). The prior probabilities P(s, a) are obtained
+25
+
+Algorithm 3 TD-n-tuple training algorithm. Input: see Algorithm 2. Output: Θ: trained
+n-tuple network parameters.
+1: function TDNTUPLETRAIN(pmax, M, Etrain, c, Rpos)
+2:
+Θ ← INITIALIZENETWORKPARAMETERS
+3:
+INITIALIZETCLPARAMETERS
+▷ Set TCL-accumulators Ni = Ai = 0, αi = 1 ∀i
+4:
+for m = 1, . . . , M do
+5:
+Perform one m-iteration of Algorithm 2 with learning rates ααi instead of α
+6:
+Ni ← Ni + ∆θi and Ai ← Ai + |∆θi|
+▷ Update TCL-accumulators
+7:
+▷ where ∆θi is the last term in Eq. (20)
+8:
+αi ← |Ni|/Ai
+∀i with Ai ̸= 0
+9:
+return Θ
+Algorithm 4 Evaluation algorithm with MCTS solver. Input: trained n-tuple network jΘ,
+p: number of scrambling twists, B: batch size, Eeval: maximum episode length during
+evaluation, I: number of MCTS-iterations, cPUCT : relative weight for U(s, a) in Eq. (24),
+dmax: maximum MCTS tree depth. Output: solved rate.
+1: function TDNTUPLEEVAL(jΘ, p, B, Eeval, I, cPUCT , dmax)
+2:
+X ←GENERATESCRAMBLEDCUBES(B, p)
+▷ B scrambled cubes
+3:
+Csolved ← 0
+4:
+for xi ∈ X do
+5:
+s ← xi
+6:
+for k = 1, . . . , Eeval do
+7:
+T ← PERFORMMCTSSEARCH(s, I, cPUCT , dmax, jΘ)
+8:
+a ← SELECTMOSTVISITEDACTION
+9:
+s ← f(s, a)
+10:
+if (s = s∗) then
+11:
+Csolved ← Csolved + 1
+12:
+break
+▷ break out of k-loop
+13:
+return Csolved/B
+▷ percentage solved
+by sending the trained network’s values of all follow-up states s′ = f(s, a) with a ∈ A(s)
+through a softmax function (see Sec. 3).16
+Once an unexpanded leaf node sℓ is reached, the node is expanded by initializing
+its accumulators: W(s, a) = N(s, a) = 0 and P(s, a) = ps′ where ps′ is the softmax-
+squashed output jΘ(s′) of our n-tuple network for each state s′ = f(s, a). The value of
+the node is the network output of the best state jΘ(sbest) = maxs′ jΘ(s′) and this value is
+backpropagated up the tree.
+More details on our MCTS wrapper can be found in Scheiermann and Konen (2022).
+16Note that the prior probabilities and the MCTS iteration are only needed at test time, so that we – different
+to AlphaZero – do not need MCTS during self-play training.
+26
+
+5.2.3
+Method Summary
+We summarize the different ingredients of our n-tuple-based TD learning method in Algo-
+rithm 3 (training) and Algorithm 4 (evaluation).
+In line 5 of Algorithm 3 we perform one m-iteration of Algorithm 2 which does an update
+step for weight vector Θ, see Eq. (20). All weights of activated n-tuple entries get a weight
+change ∆θi equal to the last term in Eq. (20) where the global α is replaced by ααi.
+Line 2 in Algorithm 4 generates a set X of B scrambled cube states. Line 7 builds for
+each xi ∈ X an MCTS tree (see Sec. 5.2.2) starting from root node xi and line 8 selects
+the most visited action of the root node. If the goal state s∗ is not found during Eeval k-loop
+trials, this xi is considered as not being solved.
+6
+Results
+6.1
+Experimental setup
+We use for all our GBG experiments the same RL method based on n-tuple systems and
+TCL. Only its hyperparameters are tuned to the specific game, as shown below. We refer
+to this method/agent as TCL-base whenever it alone is used for game playing. If we wrap
+such an agent by an MCTS wrapper with a given number of iterations, then we refer to this
+as TCL-wrap.
+We investigate two variants of Rubik’s Cube: 2x2x2 and 3x3x3. We trained all TCL
+agents by presenting them M = 3 000 000 cubes scrambled with p random twists, where
+p is chosen uniformly at random from {1, . . . , pmax}. Here, pmax = 13 [16] for 2x2x2 and
+pmax = 9 [13] for 3x3x3, where the first number is for HTM, while the second number
+in square brackets is for QTM. With these pmax cube twists we cover the complete cube
+space for 2x2x2, where God’s number (Sec. 2.2) is known to be 11 [14]. But we cover only
+a small subset in the 3x3x3 case, where God’s number is known to be 20 [26] (Rokicki
+et al., 2014).17 We train 3 agents for each cube variant { 2x2x2, 3x3x3 } × { HTM, QTM }
+to assess the variability of training.
+The hyperparameters of the agent for each cube variant were found by manual fine-
+tuning. For brevity, we defer the exact explanation and setting of all parameters to Ap-
+pendix C.
+We evaluate the trained agents for each p on 200 scrambled cubes that are created by
+applying the given number p of random scrambling twists to a solved cube. The agent now
+tries to solve each scrambled cube. A cube is said to be unsolved during evaluation if the
+agent cannot reach the solved cube in Eeval = 50 steps.18
+17We limit ourselves to pmax = 9 [13] in the 3x3x3 HTM [QTM ] case, because our network has not enough
+capacity to learn all states of the 3x3x3 Rubik’s cube. Experiments with higher twist numbers during training
+did not improve the solved-rates.
+18During training, we use lower maximum episode lengths Etrain (see Appendix C) than Eeval = 50 in
+order to reduce computation time (in the beginning, many episodes cannot be solved, and 50 would waste a
+lot of computation time). But Etrain is always at least pmax + 3 in order to ensure that the agent has a fair
+chance to solve the cube and collect the reward.
+27
+
+HTM
+0%
+25%
+50%
+75%
+100%
+1
+3
+5
+7
+9
+11
+13
+scrambling twists
+percentage solved
+cubeWidth
+2x2x2
+3x3x3
+iterMWrap
+0
+100
+800
+QTM
+0%
+25%
+50%
+75%
+100%
+1
+3
+5
+7
+9
+11
+13
+15
+scrambling twists
+percentage solved
+cubeWidth
+2x2x2
+3x3x3
+iterMWrap
+0
+100
+800
+Figure 12: Percentage of solved cubes as a function of scrambling twists p for the trained
+TD-N-tuple agent wrapped by MCTS wrapper with different numbers of iterations. The red
+curves are TCL-base without wrapper, the other colors show different forms of TCL-wrap.
+Twist type is HTM (left) and QTM (right). Each point is the average of 3 independently trained
+agents.
+6.2
+Cube Solving with MCTS Wrapper, without Symmetries
+The trained TD-N-tuple agents learn to solve the cubes to some extent, as the red curves
+TCL-base in Fig. 12 show, but they are in many cases (i.e. p > pmax/2) far from being
+perfect. These are the results from training each agent for 3 million episodes, but the
+results would not change considerably, if 10 million training episodes were used.
+Scheiermann and Konen (2022) have shown, that the performance of agents, namely
+TD-N-tuple agents, is largely improved, if the trained agents are wrapped during test, play
+and evaluation by an MCTS wrapper. This holds for Rubik’s cube as well, as Fig. 12 shows:
+For the 2x2x2 cube, the non-wrapped agent TCL-base (red curve) is already quite good,
+but with wrapping it becomes almost perfect. For the 3x3x3 cube, the red curves are not
+satisfactorily: the solved-rates are below 20% for p = 9 [13] in the HTM [QTM ] case. But
+at least MCTS wrapping boosts the solved-rates by a factor of 3 [QTM: from 16% to 48%]
+or 4.5 [HTM: from 10% to 45%].
+All these results are without incorporating symmetries.
+How symmetries affect the
+solved-rates will be investigated in Sec. 6.4. But before this, we look in the next section at
+the number of symmetries that effectively exist in a cube state.
+6.3
+Number of Symmetric States
+Not every cube state has 24 truly different symmetric states (24 = number of color sym-
+metries). For example in the solved cube, all color-symmetric states are the same (after
+normalization). Thus, we have here only one truly different symmetric state.
+However, we show in this section that for the majority of cube states the number of
+truly different symmetric states is close to 24. Two states are truly different if they are
+not the same after the normalizing operation. We generate a cube state by applying p
+random scrambling twists to the default cube. Now we apply all 24 color transformations
+(Sec. 2.4.3) to it and count the truly different states. The results are shown in Fig. 13 for
+28
+
+twistType: HTM
+twistType: QTM
+0
+4
+8
+12
+16
+0
+4
+8
+12
+16
+0
+5
+10
+15
+20
+25
+scrambling twists
+NsymmetricStates
+cubeWidth
+2x2x2
+3x3x3
+Figure 13: Count of truly different symmetric states for cube states generated by p random
+scrambling twists. Each point is an average over 500 such states.
+both cube sizes and both twist types. For the 3x3x3 cube, the number of states quickly (for
+p > 5) approaches the maximum N = 24, while for the 2x2x2 cube it is a bit slower: p > 4
+or p > 8 is needed to surpass N = 20.
+As a consequence, it makes sense to use 16 or even 24 symmetries when training
+and evaluating cube agents. Especially for scrambled states with higher p, the 24 color
+transformations used to construct symmetric states will usually lead to 24 different states.
+6.4
+The Benefit of Symmetries
+In order to investigate the benefits of symmetries, we first train a TCL agent with dif-
+ferent numbers of symmetries. As described in Sec. 2.5, we select in each step nSym
+= 0, 8, 16, 24 symmetric states. Which symmetric states are chosen is selected randomly.
+Symmetries are used (a) to update the weights for each symmetric state and (b) to build
+with Eq. (18) a smoothed value function which is used to decide about the next action dur-
+ing training. For 0, 8, 16, 24 symmetries, we train 3 agents each (3x3x3 cube, STICKER2,
+QTM). The 3 agents differ due to their differently created random-walk n-tuple sets.
+Fig. 14 shows the learning curves for different nSym = 0, 8, 16, 24. It is found that agents
+with nSym > 0 learn faster and achieve a higher asymptotic solved rate.
+Next, we evaluate each of the trained agents by trying to solve for each p ∈ {1, . . . , 15}
+(scrambling twists) 200 different scrambled cubes. During evaluation, we use again the
+same nSym as in training to form a smoothed value function. We compare in Fig. 15 different
+29
+
+50%
+60%
+70%
+80%
+90%
+0e+00
+1e+06
+2e+06
+3e+06
+episodes
+percentage solved
+nSym
+24
+16
+8
+0
+Figure 14: Learning curves for different numbers nSym = 0, 8, 16, 24 of symmetries. Shown is
+the solved rate of (3x3x3, QTM) cubes. The solved rate is the average over all twist numbers
+p = 1, . . . , 13 with 200 testing cubes for each p and over 3 agents with different random-walk
+n-tuple sets.
+symmetry results, both without wrapping (TCL-base, red curves) and with MCTS-wrapped
+agents using 100 (green) or 800 (blue) iterations. It is clearly visible that MCTS wrapping
+has a large effect, as it was also the case in Fig 12. But in addition to that, the use of
+symmetries leads for each agent, wrapped or not, to a substantial increase in solved-rates
+(a surplus of 10-20%). It is remarkable, that even for p=14 or 15 a solved rate above or
+near 50% can be reached19 by the combination (nSym=16, 800 MCTS iterations).
+Surprisingly, it seems that with wrapping it is only important whether we use symme-
+tries, not how many, since the difference between nSym = 8, 16, 24 is only marginal. For
+800 MCTS iterations, the solved rate for nSym = 24 is in most cases even smaller than that
+for nSym = 8, 16. This is surprising because it would have been expected that also with
+wrapping a larger nSym should lead to a smoother value function and thus should in theory
+produce larger solved rates. – Note that this is not a contradiction to Fig. 14, because the
+learning curves were obtained without wrapping and the red TCL-base curves in Fig. 15
+(again without wrapping) show the same positive trend with increasing nSym20. The red
+curves in Fig. 15 show approximately the same average solved rates as the asymptotic
+values in Fig. 14.
+6.5
+Computational Costs
+Table 9 shows the computational costs when training and testing with symmetries. All
+computations were done on a single CPU Intel i7-9850H @ 2.60GHz. If we subtract the
+19p is above pmax=13, the maximum twist number used during training.
+20i.e. nSym= 24 is for every p clearly better than nSym= 16
+30
+
+QTM
+3x3x3
+0%
+25%
+50%
+75%
+100%
+1
+3
+5
+7
+9
+11
+13
+15
+scrambling twists
+percentage solved
+nSym
+0
+8
+16
+24
+iterMWrap
+0
+100
+800
+Figure 15: With symmetries: Percentage of solved cubes (3x3x3, QTM) as a function of
+scrambling twists p for TD-N-tuple agents trained and evaluated with different numbers of
+symmetries nSym and wrapped by MCTS wrappers with different iterations. The red curves
+are TCL-base (without wrapper), the other colors show different forms of TCL-wrap. The
+solved rates are the average over 200 testing cubes for each p and over 3 agents with differ-
+ent random-walk n-tuple sets.
+computational costs for nsym= 0, computation time increases more or less linearly with
+iter and roughly linearly with nSym. Computation times for nSym= 24 are approximately
+10x larger than those for nSym= 0.
+Computation times are dependent on the solved rate: If a cube with p = 13 is solved,
+the episode takes normally 12-15 steps. If the cube is not solved, the episode needs 50
+steps, i.e. a factor of 3-4 more. Thus, the numbers in Table 9 should be taken only as
+rough indication of the trend.
+Bottom line: Training time through symmetries increases by a factor of 13/0.5 = 26
+(nSym= 24) and testing time increases through 800 MCTS iterations by a factor of about
+3130/8 ≈ 400.
+Training with symmetries takes between 5.4h and 13h on a normal CPU, depending
+on the number of symmetries. This is much less than the 44h on a 32-core server with 3
+GPUs that were used by McAleer et al. (2019). But it also does not reach the same quality
+as McAleer et al. (2019).
+31
+
+Table 9: Computation times with symmetries. All numbers are for 3x3x3 cube, STICKER2
+and QTM. Training: 3 million self-play episodes, w/o MCTS in the training loop. Testing: 200
+scrambled cubes with p = 13, agents wrapped by MCTS wrapper with iter iterations.
+nSym
+training
+testing
+[hours]
+[seconds]
+iter
+0
+100
+400
+800
+0
+0.5
+0.5
+48
+196
+390
+8
+5.4
+4.0
+241
+877
+1400
+16
+9.5
+7.3
+464
+1380
+2330
+24
+13.0
+8.0
+550
+1760
+3130
+7
+Related Work
+Ernö Rubik invented Rubik’s cube in 1974. Rubik’s cube has gained worldwide popularity
+with many human-oriented algorithms being developed to solve the cube from arbitrary
+scrambled start states. By ’human-oriented’ we mean algorithms that are simple to mem-
+orize for humans. They usually will find long, suboptimal solutions. For a long time it was
+an unsolved question what is the minimal number of moves (God’s Number) needed to
+solve any given cube state. The early work of Thistlethwaite (1981) put an upper bound on
+this number with his 52-move algorithm. This was one of the first works to systematically
+use group theory as an aid to solve Rubik’s cube. Later, several authors have gradually
+reduced the upper bound 52 (Joyner, 2014), until Rokicki et al. (2014) could prove in 2014
+for the 3x3x3 cube that God’s Number is 20 in HTM and 26 in QTM.
+Computer algorithms to solve Rubik’s cube rely often on hand-engineered features and
+group theory. One popular solver for Rubik’s cube is the two-phase algorithm of Kociemba
+(2015). A variant of A∗ heuristic search was used by Korf (1991), along with a pattern
+database heuristic, to find the shortest possible solutions.
+The problem of letting a computer learn to solve Rubik’s cube turned out to be much
+harder: Irpan (2016) experimented with different neural net baseline architectures (LSTM
+gave for him reportedly best results) and tried to boost them with AdaBoost. However, he
+had only for scrambling twist ≤ 7 solved rates of better than 50% and the baseline turned
+out to be better than the boosted variants. Brunetto and Trunda (2017) found somewhat
+better results with a DNN, they could solve cube states with 18 twists with a rate above
+50%. But they did not learn from scratch because they used an optimal solver based
+on Kociemba (2015) to generate training examples for the DNN. Smith et al. (2016) tried
+to learn Rubik’s cube by genetic programming. However, their learned solver could only
+reliably solve cubes with up to 5 scrambling twists.
+A breakthrough in learning to solve Rubik’s cube are the works of McAleer et al. (2018,
+2019) and Agostinelli et al. (2019): With Autodidactic Iteration (ADI) and Deep Approxi-
+mate Value Iteration (DAVI) they were able to learn from scratch to solve Rubik’s cube in
+QTM for arbitrary scrambling twists. Their method has been explained in detail already
+in Sec. 5.1, so we highlight here only their important findings: McAleer et al. (2019) only
+needs to inspect less than 4000 cubes with its trained network DeepCube when solving
+32
+
+for a particular cube, while the optimal solver of Korf (1991) inspects 122 billion different
+nodes, so Korf’s method is much slower.
+Agostinelli et al. (2019) extended the work of McAleer et al. (2019) by replacing the
+MCTS solver with a batch-weighted A∗ solver which is found to produce shorter solution
+paths and have shorter run times. At the same time, Agostinelli et al. (2019) applied their
+agent DeepCubeA successfully to other puzzles like LightsOut, Sokoban, and the 15-, 24-,
+35- and 48-puzzle21. DeepCubeA could solve all of them.
+The deep network used by McAleer et al. (2019) and Agostinelli et al. (2019) were
+trained without human knowledge or supervised input from computerized solvers. The
+network of McAleer et al. (2019) had over 12 million weights and was trained for 44 hours
+on a 32-core server with 3 GPUs. The network of McAleer et al. (2019) has seen 8 billion
+cubes during training. – Our approach started from scratch as well. It required much less
+computational effort (e.g. 5.4h training time on a single standard CPU for nSym=8, see
+Table 9). It can solve the 2x2x2 cube completely, but the 3x3x3 cube only partly (up to 15
+scrambling twists). Each trained agent for the 3x3x3 cube has seen 48 million scrambled
+cubes22 during training.
+8
+Summary and Outlook
+We have presented new work on how to solve Rubik’s cube with n-tuple systems, reinforce-
+ment learning and an MCTS solver. The main ideas were already presented in Scheier-
+mann and Konen (2022) but only for HTM and up to p = 9 twists. Here we extended
+this work to QTM as well and presented all the details of cube representation and n-tuple
+learning algorithms necessary to reproduce our Rubik’s cube results. As a new aspect,
+we added cube symmetries and studied their effect on solution quality. We found that the
+use of symmetries boosts the solved rates by 10-20%. Based on this, we could increase
+for QTM the number of scrambling twists where at least 45% of the cubes are solved from
+p = 13 without symmetries to p = 15 with symmetries.
+We cannot solve the 3x3x3 cube completely, as McAleer et al. (2019) and Agostinelli
+et al. (2019) do. But our solution is much less computational demanding than their ap-
+proach.
+Further work might be to look into larger or differently structured n-tuple systems, per-
+haps utilizing the staging principle that Ja´skowski (2018) used to produce world-record
+results in the game 2048.
+21a set of 15, 24, ... numbers has to be ordered on a 4 × 4, 5 × 5, ... square with one empty field
+223 · 106 × 16 = training episodes × episode length Etrain. This is an upper bound: some episodes may
+have shorter length, but each unsolved episode has length Etrain.
+33
+
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+
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+Sep-01-2022. 32
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+retrieved Aug-17-2022. 7
+Wikipedia.
+Rubik’s Cube, 2022b.
+URL https://en.wikipedia.org/wiki/Rubik’s_
+Cube. retrieved Aug-17-2022. 7
+36
+
+Appendix
+A
+Calculating sloc from fcol
+Given the face colors fc (Eq. (3)) of a transformed cube, how can we calculate the trans-
+formed sticker locations sℓ (Eq. (4))?
+This problem seems ill-posed at first sight, because a certain face color, e.g. white,
+appears multiple times in fc and it is not possible to tell from the appearance of white alone
+to which sticker location sℓ it corresponds. But with a little more effort, i.e. by looking at the
+neighbors of the white sticker, we can solve the problem, as we show in the following.
+A.1
+2x2x2 cube
+All cubies of the 2x2x2 cube are corner cubies. We track for each cubie exactly one sticker.
+This can be for example the set
+B = {0, 1, 2, 3, 12, 13, 14, 15}
+of 8 stickers, which is the same as the set of tracked stickers shown in Fig. 11.
+For each s ∈ B:
+1. Build the cubie that contains s as the first sticker.23
+2. Locate the cubie in fc. That is, find a location in fc with the same color as the 1st
+cubie face. If found, check if the neighbor to the right24 has the color of the 2nd cubie
+face. If yes, check if its neighbor to the right has the color of the 3rd cubie face. If
+yes, we have located the cubie in fc and we return it, i.e. its three sticker locations
+C = [a, b, c].
+3. Having located the cubie, we can infer three elements of sℓ:
+sℓ[s]
+=
+C[0]
+(26)
+sℓ[R[s]]
+=
+C[1]
+(27)
+sℓ[R[R[s]]]
+=
+C[2]
+(28)
+Here R[s] is the right neighbor of sticker s. R[R[s]]] is the left neighbor.
+In total, we have located 8 × 3 = 24 stickers, i.e. the whole transformation for sℓ.25
+23We know for example from looking at the default cube in Fig. 11 that sticker s = 0 is part of the 0-8-4-cubie.
+24By neighbor to the right we mean the next sticker when we march in clockwise orientation around the
+actual cubie.
+25The relevant GBG source code is in CubeState.locate and CubeState2x2.apply_sloc_slow.
+37
+
+A.2
+3x3x3 cube
+The 3x3x3 cube has 8 corner cubies and 12 edge cubies. We track for each cubie exactly
+one sticker. This can be for the corners the set
+B = {0, 2, 4, 6, 24, 26, 28, 30}
+and for the edges the set
+E = {1, 3, 5, 7, 25, 27, 29, 31, 11, 15, 21, 33}.
+We do for the corner set B the same as we did for the 2x2x2 cube.
+For each element s ∈ E of the edge set:
+1. Build the edge cubie cE that contains s as the first sticker.
+2. Locate the cubie in fc. That is, find an edge location in fc with the same color as the
+1st cubie face. If found, check if the other sticker of that cubie has the same color as
+the other sticker of cE. If yes, we have located the edge cubie in fc and we return it,
+i.e. its two stickers C = [a, b].
+3. Having located the cubie, we can infer two elements of sℓ:
+sℓ[s]
+=
+C[0]
+(29)
+sℓ[O[s]]
+=
+C[1]
+(30)
+Here O[s] is the other sticker of the edge cubie that has sticker s as first sticker.
+In total, we have located
+8 × 3 + 12 × 2 = 48
+stickers, i.e. the whole transformation for sℓ.26
+B
+N-Tuple Representations for the 3x3x3 Cube
+In this appendix we describe the n-tuple representations of the cube, analogously to the
+2x2x2 cube Sec. 4, but now for the 3x3x3 cube.
+B.1
+CUBESTATE
+A natural way to translate the cube state into a board is to use the flattened representation
+of Fig. 4 as the board and extract from it the 48-element vector b, according to the given
+numbering. The kth element bk represents a certain cubie face location and gets a number
+from {0, . . . , 5} according to its current face color fc. The solved cube is for example
+represented by b = [00000000 11111111 . . . 55555555].
+This representation CUBESTATE is what the BoardVecType CUBESTATE in our GBG-
+implementation means: Each board vector is a copy of fcol, the face colors of all cubie
+faces. An upper bound of possible combinations for b is 648 = 2.2 · 1032. This is much
+larger than the true number of distinct states (Sec. 2.2.2) which is 4.3 · 1019.
+26The relevant GBG source code is in CubeState.locate, CubeState3x3.locate_edge and CubeState3x3.apply_sloc_slow.
+38
+
+Table 10: The correspondence edge location ↔ STICKER2 for the solved cube. The yellow
+colored cells show the location of the 12 edge stickers that we track.
+3x3x3
+location
+1 3 5
+7
+9 11 13 15
+17 19 21 23
+25 27 29 31
+33 35 37 39
+41 43 45 47
+STICKER2
+edge
+A B C D
+D
+G
+K
+E
+E
+J
+F
+A
+I
+J
+K
+L
+H
+B
+F
+I
+L
+G
+C
+H
+face ID
+1 1 1
+1
+2
+2
+2
+2
+1
+1
+1
+1
+1
+1
+1
+1
+2
+2
+2
+2
+1
+1
+1
+1
+B.2
+STICKER
+McAleer et al. (2019) had the interesting idea for the 3x3x3 cube that 20 stickers (cubie
+faces) are enough. To characterize the 3x3x3 cube, we need according to McAleer et al.
+(2019) only one (not 2 or 3) sticker for every of the 20 cubies, as shown in Fig. 10. This
+is because the location of one sticker uniquely defines the location and orientation of that
+cubie. We name this representation STICKER in GBG.
+We track the 4 top corner stickers 0,2,4,6 plus the 4 bottom corner stickers 24,26,28,30
+plus one sticker for each of the 12 edge stickes as shown in Fig. 10, in total 20 stickers and
+ignore the 28 other stickers.
+How to lay out this representation as a board? – McAleer et al. (2019) create a rect-
+angular one-hot-encoding board with 20 × 24 = 480 cells (20 rows for the stickers and
+24 columns for the locations27) carrying only 0’s and 1’s. This is fine for the approach of
+McAleer et al. (2019), where they use this board as input for a DNN, but not so nice for
+n-tuples. Without constraints, such a board amounts to 2480 ≈ 10145 combinations, which
+is unpleasantly large (much larger than in CUBESTATE).28
+Another possibility to lay out the board: Specify 20 board cells (the stickers) with 24
+position values each. This amounts to 2420 = 4.0 · 1027 combinations.
+B.3
+STICKER2
+Analogously to Sec. 4.3, we represent the 24 corner locations and 24 edge locations as:
+corner location = (corner cubie, face ID),
+edge location = (edge cubie, face ID).
+That is, each corner location is represented by a corner cubie a,b,c,d,e,f,g,h and by a face
+ID 1,2,3. Table 7 shows the explicit numbering in this new representation. Additionally,
+each edge location is represented by an edge cubie A,B,C,D,E,F,G,H,I,J,K,L29 and by a
+face ID 1,2. Convention for face ID numbering of edge cubies: For top- and bottom-layer
+edge cubies, it is 1 for U and D stickers, 2 else. The face ID for middle-layer edge cubies is
+1 for F and B stickers, 2 else. Table 10 shows the explicit numbering in this representation.
+The corresponding board consists of 8 + 8 + 12 +12 = 40 cells shown in Table 11.
+The 8 cell pairs in the first two rows code the locations of the tracked corner stickers
+278 · 3 for the corner stickers and 12 · 2 for the edge stickers
+28McAleer et al. (2019) do not need a weight for every of the 2480 possible states, as the n-tuple network
+would need. Instead they need only 480 · 4096 = 2 · 106 weights to the first hidden layer having 4096 neurons.
+294 U-stickers, 4 D-sticker, 4 middle-layer stickers (2F, 2B)
+39
+
+0,2,4,6,24,26,28,30, see Table 7 in Sec. 4.3. The 12 cell pairs in the last two rows code the
+location of the tracked edge stickers 1,3,5,7,17,21,43,47,25,27,29,31, see Table 10. This
+n-tuple coding requires tuple cells with varying number of position values and leads to
+88 · 38 · 1212 · 212 = 4.0 · 1027
+combinations in representation STICKER2.30
+Table 11: STICKER2 board representation for the default 3x3x3 cube. For the BoardVector,
+cells are numbered row-by-row from 0 to 39.
+corner
+a
+b
+c
+d
+e
+f
+g
+h
+8 positions
+face ID
+1
+1
+1
+1
+1
+1
+1
+1
+3 positions
+edge
+A
+B
+C
+D
+E
+F
+G
+H
+I
+J
+K
+L
+12 positions
+face ID
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+2 positions
+B.4
+Adjacency Sets
+To create n-tuples by random walk, we need to define adjacency sets (sets of neighbors)
+for every board cell k.
+For CUBESTATE, the board is the flattened representation of the 3x3x3 cube (Fig. 4).
+The adjacency set is defined as the 4-point neighborhood, where two stickers are neigh-
+bors if they are neighbors (share a common edge) on the cube.
+For STICKER2, the board consists of 40 cells shown in Table 11. Since it matters for
+the corner stickers mostly where the other corner stickers are and for the edge stickers
+mostly where the other edge stickers are, it is reasonable to form two adjacency subsets
+S1 = {00, . . . , 15} and S2 = {16, . . . , 39} and to define the adjacency set
+Adj(k) = Si \ {k}
+for each k ∈ Si, i = 1, 2.
+C
+Hyperparameters
+In this appendix we list all parameter settings for the GBG agents used in this paper. Pa-
+rameters were manually tuned with two goals in mind: (a) to reach high-quality results and
+(b) to reach stable (robust) performance when conducting multiple training runs with differ-
+ent random seeds. The agents listed further down are the best-so-far agents found (best
+among all agents that learn from scratch by self-play).
+The detailed meaning of RL parameters is explained in Konen and Bagheri (2021):
+30This is, by the way, identical to (8·3)8 ·(12·2)12 = 24(8+12) = 2420 = 4.0·1027, the same number we had
+above in the second mode of STICKER. But STICKER2 has the advantage that the combinations are spread
+over more board cells (40) than in STICKER (20). By having more board cells with fewer position values, the
+n-tuples can better represent the relationships between cube states.
+40
+
+• Algorithms 2, 5 and 7 in Konen and Bagheri (2021) explain parameters α (learning
+rate), γ (discount factor), ϵ (exploration rate) and output sigmoid σ (either identity or
+tanh).
+• Appendix A.3 explains our eligibility method, parameters are: eligibility trace factor λ,
+horizon cut ch, eligibility trace type ET (normal) or RESET (reset on random move).
+If not otherwise stated, we use in this paper λ = 0 (no eligibility traces). For λ = 0,
+horizon cut ch and eligibity trace type are irrelevant. If λ > 0, their defaults ch = 0.1
+and trace type ET apply.
+• Appendix A.5 explains our TCL method (also summarized in Sec. 5.2.1). Parameters
+of TCL are: TC-Init (initialization constant for counters), TC transfer function (TC-id
+or TC-EXP), β (exponential factor in case of TC-EXP), TC accumulation type (delta
+or recommended weight-change).
+Another branch of our algorithm is the MCTS wrapper, which can be used to wrap
+TD-N-tuple agents during evaluation and testing. MCTS wrapping is briefly explained in
+Sec. 5.2.2. The precise algorithm for MCTS wrapping is explained in detail in (Scheiermann
+and Konen, 2022, Sec. II-B).31 Parameters of MCTS are:
+• cPUCT : relative weight for the prior probabilities of the wrapped agent in relation to
+the value that the wrapper estimates
+• dmax: maximum depth of the MCTS tree, if -1: no maximum depth
+• UseSoftMax: boolean, whether to use SoftMax normalization for the priors or not
+• UseLastMCTS: boolean, whether to re-use the MCTS from the previous move within
+an episode or not
+Further parameter explanations:
+• Sec. 4 in this document explains n-tuples, parameters are: number of n-tuples, length
+of n-tuples, and n-tuple creation mode (fixed, random walk, random points).
+• Sec. 2.5 in this document explains symmetries. If parameter nSym = 0, do not use
+symmetries. If nSym > 0, use this number nSym of symmetries. In the Rubik’s cube
+case, nSym is a number between 0 and 24.
+• LearnFromRM: whether to learn from random moves or not. (Does not apply here,
+because we use in Rubiks’s cube always ϵ = 0, i.e. we have no random moves.)
+• ChooseStart-01: whether to start episodes from different 1-ply start states or always
+from the default start state. (Does not apply here, because we start in Rubik’s cube
+never from the default cube, but always from the p-twisted cube.)
+31As (Scheiermann and Konen, 2022, Sec. IV-E) shows, the MCTS wrapper may be used as well during
+training, but due to large computation times needed for this, we do not follow that route in this paper.
+41
+
+• Etrain: maximum episode length during training, if -1: no maximum length.
+• Eeval: maximum episode length during evaluation and play, if -1: no maximum length.
+All agents were trained with no MCTS wrapper inside the training loop. The hyper-
+parameters of the agent for each cube variant were found by manual fine-tuning. See
+also (Konen, 2022).
+In the following, we list the precise settings for all agents used in this paper. If not stated
+otherwise, these common settings apply to all agents: sigmoid σ = id, LearnFromRM =
+false, ChooseStart-01 = false. Wrapper settings during test and evaluation: MCTS wrapper
+with cPUCT = 1.0, dmax = 50, UseSoftMax = true, UseLastMCTS = true.
+Common parameters of Algorithm 2 in Sec. 5.2 are: cost-to-go c = −0.1 and positive
+reward Rpos = 1.0.
+The parameters for training without symmetries (nSym = 0) in Sec. 6.2 are:
+• 2x2x2 cube, HTM: α = 0.25, γ = 1.0, ϵ = 0.0, λ = 0.0, no output sigmoid. N-tuples:
+60 7-tuples created by random walk.
+TCL activated with transfer function TC-id,
+TC-Init= 10−4 and rec-weight-change accumulation. 3,000,000 training episodes.
+pmax = 13, Etrain = 16, Eeval = 50.
+Agent filename in GBG: 2x2x2_STICKER2_AT/TCL4-p13-ET16-3000k-60-7t-stub.agt.zip
+• 2x2x2 cube, QTM: same as 2x2x2 cube, HTM, but with pmax = 16, Etrain = 20.
+Agent filename in GBG: 2x2x2_STICKER2_QT/TCL4-p16-ET20-3000k-60-7t-stub.agt.zip
+• 3x3x3 cube, HTM: same as 2x2x2 cube, HTM, but with 120 7-tuples created by
+random walk, pmax = 9, Etrain = 13.
+Agent filename in GBG: 3x3x3_STICKER2_AT/TCL4-p9-ET13-3000k-120-7t-stub.agt.zip
+• 3x3x3 cube, QTM: same as 3x3x3 cube, HTM, but with pmax = 13, Etrain = 16.
+Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-stub.agt.zip
+The agent files given in the list above are just stubs, i.e. agents that are initialized with
+the correct parameters but not yet trained. This is because a trained agent can require up
+to 80 MB disk space, which is too much for GitHub. Instead, a user of GBG may load such
+a stub agent, train it (takes between 10-40 minutes) and save it to local disk.
+When evaluating in Sec. 6.2 the trained agents with different MCTS wrappers, we test
+in each case whether cPUCT = 1.0 or 10 is better. In most cases, cPUCT = 1.0 is better,
+but for (2x2x2, QTM, 800 iterations) and for (3x3x3, HTM, 100 iterations) cPUCT = 10.0 is
+the better choice.
+The parameters for training with symmetries (nSym = 8, 16, 24) in Sec. 6.4 are:
+• 3x3x3 cube, QTM: same as 3x3x3 cube, QTM in Sec. 6.2, but with nsym = 8, 16, 24.
+Agent filename in GBG: 3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym08-stub.agt.zip,
+3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym16-stub.agt.zip,
+3x3x3_STICKER2_QT/TCL4-p13-ET16-3000k-120-7t-nsym24-stub.agt.zip.
+42
+
+Again, the agent filenames are just stubs, i.e. agents that are initialized with the correct
+parameters but not yet trained. As above, a user of GBG may load such a stub agent, train
+it (which takes in the symmetry case between 5.4h and 13h, see Table 9) and save it to
+local disk.
+For further details and experiment shell scripts, see also the associated Papers-with-
+Code repository https://github.com/WolfgangKonen/PapersWithCodeRubiks.
+43
+

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+VANISHING OF QUARTIC AND SEXTIC TWISTS OF
+L-FUNCTIONS
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Abstract. Let E be an elliptic curve over Q.
+We conjecture
+asymptotic estimates for the number of vanishings of L(E, 1, χ) as
+χ varies over all primitive Dirichlet characters of orders 4 and 6.
+Our conjectures about these families come from conjectures about
+random unitary matrices as predicted by the philosophy of Katz-
+Sarnak. We support our conjectures with numerical evidence.
+Earlier work by David, Fearnley and Kisilevsky formulates anal-
+ogous conjectures for characters of any odd prime order. In the
+composite order case, however, we need to justify our use of random
+matrix theory heuristics by analyzing the equidistribution of the
+squares of normalized Gauss sums. Along the way we introduce the
+notion of totally order ℓ characters to quantify how quickly quartic
+and sextic Gauss sums become equidistributed. Surprisingly, the
+rate of equidistribution in the full family of quartic (sextic, resp.)
+characters is much slower than in the sub-family of totally quar-
+tic (sextic, resp.) characters. A conceptual explanation for this
+phenomenon is that the full family of order ℓ twisted elliptic curve
+L-functions, with ℓ even and composite, is a mixed family with
+both unitary and orthogonal aspects.
+1. Introduction
+Vanishings of elliptic curve L-functions at the value s = 1 (normalized
+so that the functional equation relates s and 2−s) is central to a great
+deal of modern number theory. For instance, if an L-function associated
+to an elliptic curve vanishes at s = 1, then the BSD conjecture predicts
+that the curve will have infinitely many rational points.
+Additionally, statistical questions about how often L-functions within
+a family vanish at the central value have also been of broad interest.
+For example, it is expected (as first conjectured by Chowla [Cho87])
+that, for all primitive Dirichlet characters χ, we have L(χ, 1/2) ̸= 0.
+A fruitful way of studying such questions has been to model L-functions
+using random matrices.
+For example, in [CKRS00] Conrey, Keat-
+ing, Rubinstein and Snaith consider the family of twisted L-functions
+1
+arXiv:2301.05329v1  [math.NT]  12 Jan 2023
+
+2
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+L(f, s, χd) associated to a modular form f of weight k and quadratic
+characters χd. They show that the random matrix theory model pre-
+dicts that infinitely many values L(f, s, χd) are zero when the weight
+of f is 2 or 4, but that only finitely many of the values are zero when
+the weight is at least 6.
+Another example, due to David, Fearnley and Kisilevsky [DFK04,
+DFK07], instead uses the random matrix model to give conjectural
+asymptotics for the number of vanishings of elliptic curve L-functions
+twisted by families of Dirichlet characters of a fixed order. In particu-
+lar, they predict that for an elliptic curve E, the values L(E, 1, χ) are
+zero infinitely often if χ has order 3 or 5, but for characters χ with a
+fixed prime order ℓ ≥ 7, only finitely many values L(E, 1, χ) are zero.
+In recent work, inspired by the conjectures of [DFK04, DFK07], Mazur
+and Rubin [MR21] use statistical properties of modular symbols to
+heuristically estimate the probability that L(E, 1, χ) vanishes. Their
+Conjecture 11.1 implies that for an elliptic curve E over Q, there
+should be only finitely many characters χ of a fixed order ℓ such that
+L(E, 1, χ) = 0 and ϕ(ℓ) > 4. This further implies the following: Let E
+be an elliptic curve over Q and let be F/Q an infinite abelian exten-
+sion such that Gal(F/Q) has only finitely many characters of orders 2,
+3 and 5. Then E(F) is finitely generated. Finally, for an elliptic curve
+E defined over Q, their Proposition 3.2 relates the (order of) vanishing
+of L(E, 1, χ) to the growth in rank of E over a finite abelian extension
+F/Q. In particular, if BSD holds for E over both Q and F, then
+rank(E(F)) = rank(E(Q)) +
+�
+χ:Gal(F/Q)→C×
+ords=1L(E, s, χ).
+1.1. Notation and statement of the Main Conjecture. We fix
+the following notation. See Definition 3.1 for the definition of totally
+order ℓ characters but, roughly speaking, these are order ℓ characters
+that, when factored, have all their factors also of order ℓ. Set
+Ψℓ = {primitive Dirichlet characters χ of order ℓ}
+Ψtot
+ℓ
+= {χ ∈ Ψℓ that are totally order ℓ}
+Ψ′
+ℓ = {χ ∈ Ψℓ with cond(χ) prime}.
+Note that Ψ′
+ℓ ⊆ Ψtot
+ℓ
+⊆ Ψℓ.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+3
+Along the way we will need to estimate the number of characters in
+each family and so we define:
+Ψℓ(X) = {χ ∈ Ψℓ : cond(χ) ≤ X}
+Ψtot
+ℓ (X) = {χ ∈ Ψtot
+ℓ
+: cond(χ) ≤ X}
+Ψ′
+ℓ(X) = {χ ∈ Ψ′
+ℓ : cond(χ) ≤ X}.
+For an elliptic curve E over Q we also define:
+FΨℓ,E = {L(E, s, χ) : χ ∈ Ψℓ}
+FΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E : χ ∈ Ψℓ(X)}.
+We also define FΨtot
+ℓ
+,E and FΨtot
+ℓ
+,E(X) analogously for Ψtot
+ℓ
+in place of
+Ψℓ; we do the same with Ψ′
+ℓ, as well. Finally, let
+VΨℓ,E(X) = {L(E, s, χ) ∈ FΨℓ,E(X) : L(E, 1, χ) = 0}
+VΨtot
+ℓ
+,E(X) = {L(E, s, χ) ∈ FΨtot
+ℓ
+,E(X) : L(E, 1, χ) = 0}
+VΨ′
+ℓ,E(X) = {L(E, s, χ) ∈ FΨ′
+ℓ,E(X) : L(E, 1, χ) = 0}.
+With this notation, we make the following conjecture.
+Conjecture 1.1. Let E be an elliptic curve. Then, there exist con-
+stants bE,4 and bE,6 so that
+|VΨ4,E(X)| ∼ bE,4X1/2 log5/4 X
+and
+|VΨ6,E(X)| ∼ bE,6X1/2 log9/4 X
+as X → ∞.
+Moreover, if we restrict only to those twists by totally quartic or totally
+sextic characters, then there exist constants btot
+E,4 and btot
+E,6 such that
+|VΨtot
+4 ,E(X)| ∼ btot
+E,4X1/2 log1/4 X
+and
+|VΨtot
+6 ,E(X)| ∼ btot
+E,6X1/2 log1/4 X
+as X → ∞.
+Finally, if we restrict only to those twists by characters of prime con-
+ductor, then there exist constants b′
+E,4 and b′
+E,6 such that
+|VΨ′
+4,E(X)| ∼ b′
+E,4X1/2 log−3/4 X
+and
+|VΨ′
+6,E(X)| ∼ b′
+E,6X1/2 log−3/4 X
+as X → ∞.
+
+4
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+In particular, we conjecture that families of elliptic curve L-functions
+twisted by quartic and sextic characters vanish infinitely often at the
+central value.
+To assist the reader in comparing the powers of log X in the above
+asymptotics, we point out here that for ℓ = 4, |Ψ4(X)| is roughly log X
+times as large as |Ψtot
+4 (X)|, which in turn is roughly log X times as
+large as |Ψ′
+4(X)|. For ℓ = 6, then |Ψ6(X)|/|Ψtot
+6 (X)| ≍ (log X)2, and
+|Ψtot
+6 (X)|/|Ψ′
+6(X)| ≍ log X. Hence, in each of the three families with
+a given value of ℓ, the proportion of vanishing twists has the same
+order of magnitude. See Proposition 3.6, Lemma 3.7, Proposition 3.8,
+and Lemma 3.9 below for asymptotics of the underlying families of
+characters.
+1.2. Outline of the paper. There are two main ingredients needed
+to be able to apply random matrix theory predictions to our families of
+twists. The first is a discretization for the central values. As described
+in Section 2.1 this can be done for curves E satisfying certain technical
+conditions as described in [WW20]. We need this discretization in order
+to approximate the probability that L(E, 1, χ) vanishes.
+The second ingredient is a proper identification of the symmetry type
+of the family, which is largely governed by the distribution of the sign of
+the functional equation within the family (see Section 4 of [CFK+05]).
+This directly leads to an investigation around the equidistribution of
+squares of Gauss sums of quartic and sextic characters, which has con-
+nections to the theory of metaplectic automorphic forms [Pat87].
+See Section 3.1 for a thorough discussion.
+It is a subtle feature that the families of twists of elliptic curve L-
+functions by the characters in Ψtot
+ℓ
+and Ψ′
+ℓ have unitary symmetry
+type, but for composite even values of ℓ, the twists by Ψℓ should be
+viewed as a mixed family. To elaborate on this point, consider the case
+that ℓ = 4, and first note that a character χ ∈ Ψ4 factors uniquely
+as a totally quartic character times a quadratic character of relatively
+prime conductors. The totally quartic family has a unitary symmetry,
+but the family of twists of an elliptic curve by quadratic characters has
+orthogonal symmetry. This tension between the totally quartic aspect
+and the quadratic aspect is what leads to the mixed symmetry type.
+The situation is analogous to the family L(E, 1 + it, χd); if t = 0 and
+d varies then one has an orthogonal family, while if d is fixed and t
+varies, then one has a unitary family. See [SY10] for more discussion
+on this family.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+5
+Another interesting feature of these families is that Ψℓ(X) is larger
+than Ψtot
+ℓ (X) by a logarithmic factor. For instance, when ℓ = 4, then
+Ψtot
+4 (X) grows linearly in X (see Proposition 3.6 below), and of course
+Ψ2(X) also grows linearly in X. Similarly to how the average size of the
+divisor function is log X, this indicates that |Ψ4(X)| grows like X log X
+(see Lemma 3.7 below).
+The rest of the paper is organized as follows.
+In the next section
+we give the necessary background and notation for L-functions and
+their central values and discuss the discretization we use in the paper.
+In the subsequent section we estimate some sums involving quartic
+and sextic characters and discuss totally quartic and sextic characters
+in more detail. In the final section, we motivate the asymptotics in
+Conjecture 1.1 and provide numerical evidence that supports them.
+Acknowledgments. We thank David Farmer and Brian Conrey for
+helpful conversations. This research was done using services provided
+by the OSG Consortium [PPK+07, SBH+09], which is supported by the
+National Science Foundation awards #2030508 and #1836650. This
+material is based upon work supported by the National Science Foun-
+dation under agreement No. DMS-2001306 (M.Y.). Any opinions, find-
+ings and conclusions or recommendations expressed in this material are
+those of the authors and do not necessarily reflect the views of the Na-
+tional Science Foundation.
+2. L-functions and central values
+Let E be an elliptic curve defined over Q of conductor NE. The L-
+function of E is given by the Euler product
+L(E, s) =
+�
+p∤NE
+�
+1 − ap
+ps +
+1
+p2s−1
+�−1 �
+p|NE
+�
+1 − ap
+ps
+�−1
+=
+�
+n≥1
+an
+ns .
+The modularity theorem [BCDT01, TW95, Wil95] implies that L(E, s)
+has an analytic continuation to all of C and satisfies the functional
+equation
+Λ(E, s) =
+� √NE
+2π
+�s
+Γ(s)L(E, s) = wEΛ(E, 2 − s)
+where the sign of the functional equation is wE = ±1 and is the eigen-
+value of the Fricke involution. Let χ be a primitive character and let
+cond(χ) be its conductor and suppose that cond(χ) is coprime to the
+
+6
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+conductor NE of the curve. The twisted L-function has Dirichlet series
+L(E, s, χ) =
+�
+n≥1
+anχ(n)
+ns
+and the functional equation (cf. [IK04, Prop. 14.20])
+Λ(E, s, χ) =
+�
+cond(χ)√NE
+2π
+�s
+Γ(s)L(E, s, χ)
+= wEχ(NE)τ(χ)2
+cond(χ)
+Λ(E, 2 − s, χ),
+(2.1)
+where τ(χ) = �
+r∈Z/mZ χ(r)e2πir/m is the Gauss sum and m = cond(χ).
+2.1. Discretization. To justify our Conjecture 1.1, we need a condi-
+tion that allows us to deduce that L(E, 1, χ) = 0, for a given E and
+χ of order ℓ. In particular, we show that L(E, 1, χ) is discretized (see
+Lemma 4.2) and so there exists a constant cE,ℓ such that |L(E, 1, χ)| <
+cE,ℓ/
+�
+cond(χ) implies L(E, 1, χ) = 0. In this section we prove the
+results necessary for the discretization.
+Let E be an elliptic curve over Q with conductor NE.
+Let χ be a
+nontrivial primitive Dirichlet character of conductor m and order ℓ.
+Set ϵ = {±1} = χ(−1) depending on whether χ is an even or odd
+character. Let Ω+(E) and Ω−(E) denote the real and imaginary periods
+of E, respectively, with Ω+(E) > 0 and Ω−(E) ∈ iR>0.
+The algebraic L-value is defined by
+(2.2)
+Lalg(E, 1, χ) := L(E, 1, χ) · m
+τ(χ)Ωϵ(E)
+= ϵ · L(E, 1, χ)τ(χ)
+Ωϵ(E)
+While it has been known for some time that algebraic L-values are
+algebraic numbers, recent work of Weirsema and Wuthrich [WW20]
+characterizes conditions on E and χ which guarantee integrality. In
+particular, under the assumption that the Manin constant c0(E) = 1,
+if the conductor m is not divisible by any prime of additive reduction for
+E, then Lalg(E, 1, χ) ∈ Z[ζℓ] is an algebraic integer [WW20, Theorem
+2]. For a given curve E, we will avoid the finitely many characters χ
+for which Lalg(E, 1, χ) fails to be integral.
+Proposition 2.1. Let χ be a primitive Dirichlet character of odd order
+ℓ and conductor m. Then
+Lalg(E, 1, χ) =
+�
+χ(NE)(ℓ+1)/2 nE(χ),
+if wE = 1,
+(ζℓ − ζ−1
+ℓ )−1 χ(NE)(ℓ+1)/2 nE(χ)
+if wE = −1,
+for some algebraic integer nE(χ) ∈ Z[ζℓ + ζ−1
+ℓ ] = Z[ζℓ] ∩ R.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+7
+Proposition 2.2. Let χ be a primitive Dirichlet character of even
+order ℓ and conductor m. Then Lalg(E, 1, χ) = kE nE(χ) where nE(χ)
+is some algebraic integer in Z[ζℓ +ζ−1
+ℓ ] = Z[ζℓ]∩R and kE is a constant
+depending only on the curve E. In particular, when wE = 1 we have
+kE =
+�
+�
+�
+�
+�
+(1 + χ(NE))
+if χ(NE) ̸= −1
+ζℓ/4
+ℓ
+,
+if 4 | ℓ and χ(NE) = −1
+(ζℓ − ζ−1
+ℓ )
+if 4 ∤ ℓ and χ(NE) = −1.
+Proof of Prop 2.1 and Prop 2.2. Since E is defined over Q, we have
+L(E, 1, χ) = L(E, 1, χ). Using the functional equation, we obtain
+Lalg(E, 1, χ) = ϵ · L(E, 1, χ)τ(χ)
+Ωϵ(E)
+= ϵ · wE χ(NE) τ(χ)τ(χ)2
+m · Ωϵ(E)
+L(E, 1, χ)
+= wE χ(NE) τ(χ)
+Ωϵ(E)
+L(E, 1, χ)
+= wEχ(NE) ϵ · τ(χ)L(E, 1, χ)
+Ωϵ(E)
+= wEχ(NE) Lalg(E, 1, χ)
+Thus Lalg(E, 1, χ) is a solution to the equation z = wEχ(NE)z. Note
+that if z1, z2 ∈ Z[ζℓ] are two distinct solutions to this equation, then
+z1/z1 = z2/z2 so that z1/z2 = z1/z2 = (z1/z2), hence z1/z2 ∈ R. Thus
+Lalg(E, 1, χ) = αz with α ∈ Z[ζℓ] ∩ R = Z[ζℓ + ζ−1
+ℓ ] and z ∈ Z[ζℓ].
+Suppose that wE = 1. When ℓ is odd, we can take z = χ(NE)
+ℓ+1
+2 . Now
+suppose that ℓ is even. If χ(NE) ̸= −1, since χ(NE) = ζr
+ℓ for some
+1 ≤ r ≤ ℓ, we may take z = (1+χ(NE)). Indeed, we have wEχ(NE)z =
+ζr
+ℓ (1 + ζℓ−r
+ℓ
+) = ζr
+ℓ + 1 = z. If 4 | ℓ and χ(NE) = −1 = ζℓ/2
+ℓ
+, we take
+z = ζℓ/4
+ℓ
+. Finally, if 4 ∤ ℓ and χ(NE) = −1 take z = ζℓ−ζ−1
+ℓ
+= 2i Im(ζℓ).
+When wE = −1 and ℓ is odd, we may take z = (ζℓ − ζ−1
+ℓ )−1χ(NE)
+ℓ+1
+2 .
+When ℓ is even, if χ(NE) = −1 then we may take z = ζℓ + ζ−1
+ℓ
+=
+2 Re(ζℓ), and if χ(NE) ̸= −1 then we make take z = 1 − χ(NE).
+□
+Remark 2.3. We note that for ℓ even, |kE| ≤ 2.
+It is clear that
+|ζℓ/4
+ℓ
+| = 1 and |2i Im(ζℓ)| ≤ 2.
+Observe |(1 + χ(NE)| ≤ 2, by the
+triangle inequality.
+
+8
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Note that since L(E, 1, χ) vanishes if and only if nE(χ) does, we may
+interpret the integers nE(χ) as a discretization of the special values
+L(E, 1, χ). This is similar to the case of cubic characters considered in
+[DFK04] since Q(ζ3)+ = Q, as opposed to characters of prime order ℓ ≥
+5 where further steps were needed to find an appropriate discretization
+[DFK07].
+3. Estimates for Dirichlet characters
+In this section we discuss various aspects of Dirichlet characters of
+order 4 and 6. A necessary condition for a family of L-functions to
+be modeled by the family of unitary matrices is that the signs must
+be uniformly distributed on the unit circle. From (2.1), L(E, s, χ) has
+sign wEχ(NE) τ(χ)2
+cond(χ); we will largely focus on the distribution of the
+square of the Gauss sums, viewing the extra factor χ(NE) as a minor
+perturbation. To obtain our estimates for the number of vanishings
+|VΨℓ,E(X)| (respectively, |VΨ′
+ℓ,E(X)| and |VΨtot
+ℓ
+,E(X)|) we must estimate
+the size of Ψℓ(X) (respectively, Ψ′
+ℓ(X) and Ψtot
+ℓ (X)) as well as the size
+of an associated sum. We also discuss the family of totally quartic
+and sextic characters to explain some phenomena we observed in our
+computations.
+3.1. Distributions of Gauss sums. Patterson [Pat87], building on
+work of Heath-Brown and Patterson [HBP79] on the cubic case, showed
+that the normalized Gauss sum τ(χ)/
+�
+cond(χ) is uniformly distributed
+on the circle for χ varying in each of Ψtot
+ℓ
+and Ψ′
+ℓ. This result was
+first announced in [PHH81]; see [BE81] for an excellent summary of
+this and other work related to the distributions of Gauss sums. Patter-
+son’s method moreover shows that the argument of τ(χ)χ(k) is equidis-
+tributed for any fixed nonzero integer k, and hence so is the argument
+of τ(χ)2χ(k).
+For the case of quartic and sextic characters with arbitrary conductors,
+there do not appear to be any results in the literature that imply their
+Gauss sums are uniformly distributed. In Figure 1 we see the distri-
+butions of Gauss sums of characters of orders 3 through 9 of arbitrary
+conductor up to 200000. We included characters of order 4 and 6 since
+those examples are the focus of the paper; we included characters of or-
+ders 3, 5, and 7 as consistency checks (in [DFK04, DFK07] the authors
+rely on them being uniformly distributed); and we included composite
+orders 8 and 9 to see if something similar happens in those cases as
+happens in the quartic case. In all cases but the quartic case, we see
+that the distributions of the angles of the signs appear to be uniformly
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+9
+Figure 1. Each histogram represents the distribution
+of the argument of the τ(χ)2/cond(χ) for characters of
+order 3 through 9, from top left to bottom right. Each
+histogram is made by calculating the Gauss sums of char-
+acters in Ψℓ of each conductor up to 200000.
+distributed. The quartic distribution has two obvious peaks that we
+discuss below, in Remark 3.17.
+The images in Figure 1 suggest that the family of matrices that best
+models the vanishing of L(E, 1, χ) is unitary in every case except possi-
+bly the case of quartic characters. Nevertheless, in Section 3.4 we show
+that the squares of the quartic Gauss sums are indeed equidistributed,
+despite what the data suggest. Indeed, we prove that the squares of
+the sextic and quartic Gauss sums are equidistributed, allowing us to
+apply the heuristics from random matrix theory as in Section 4.
+3.2. Totally quartic and sextic characters. Much of the back-
+ground material in this section can be found with proofs in [IR90,
+Ch. 9].
+Definition 3.1. Let χ be a primitive Dirichlet character of conductor
+q and order ℓ. For prime p, let vp be the p-adic valuation, so that
+q = �
+p pvp(q). We correspondingly factor χ = �
+p χ(p), where χ(p) has
+conductor pvp(q). We say that χ is totally order ℓ if each χp is exact
+
+5+00
+5000
+4000
+DORE
+2400
+1400
+E-
+-1
+0ADODE
+24000
+E-
+-2
+-1
+0
+1DOS
+4400
+3000
+DOZ
+1400
+2
+-1
+0
+112000
+40000
+E-
+-2
+-1 
+0
+34400
+DOSE
+DODE
+2500
+DOZ
+1500
+1400
+500
+0
+E-
+-1
+i
+2
+350000
+40000
+ADODE
+DO
+4000
+E-
+-2
+-1
+0
+i217500
+15000
+12500
+14000
+DOSr
+5000
+2500
+E-
+i210
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+order ℓ.
+By convention we also consider the trivial character to be
+totally order ℓ for every ℓ.
+3.2.1. Quartic characters. The construction of quartic characters uses
+the arithmetic in Z[i]. The ring Z[i] has class number 1, unit group
+{±1, ±i}, and discriminant −4. We say α ∈ Z[i] with (α, 2) = 1 is
+primary if α ≡ 1 (mod (1+i)3). Any odd element in Z[i] has a unique
+primary associate, which comes from the fact that the unit group in
+the ring Z[i]/(1+i)3 may be identified with {±1, ±i}. An odd prime p
+splits as p = ππ if and only if p ≡ 1 (mod 4). Given π with N(π) = p,
+define the quartic residue symbol [ α
+π] for α ∈ Z[i] with (α, π) = 1,
+by [ α
+π] ∈ {±1, ±i} and [ α
+π] ≡ α
+p−1
+4
+(mod π). The map χπ(α) = [ α
+π]
+from (Z[i]/(π))× to {±1, ±i} is a character of order 4.
+If α ∈ Z,
+then [ α
+π]2 ≡ α
+p−1
+2
+≡ ( α
+p) (mod π). Therefore, χ2
+π(α) = (α
+p), showing
+in particular that the restriction of the quartic residue symbol to Z
+defines a primitive quartic Dirichlet character of conductor p.
+Lemma 3.2. Every primitive totally quartic character of odd conductor
+is of the form χβ, where β = π1 . . . πk is a product of distinct primary
+primes, (β, 2β) = 1, and where
+(3.1)
+χβ(α) =
+�α
+β
+�
+=
+k
+�
+i=1
+� α
+πi
+�
+.
+The totally quartic primitive characters of even conductor are of the
+form χ2χβ where χ2 is one of four quartic characters of conductor 24,
+and χβ is totally quartic of odd conductor.
+Proof. We begin by classifying the quartic characters of odd prime-
+power conductor. If p ≡ 3 (mod 4), there is no quartic character of
+conductor pa, since φ(pa) = pa−1(p − 1) ̸≡ 0 (mod 4). Since φ(p) =
+p − 1, if p ≡ 1 (mod 4), there are two distinct quartic characters of
+conductor p, namely, χπ and χπ, where p = ππ. There are no primitive
+quartic characters modulo pj for j ≥ 2.
+To see this, suppose χ is
+a character of conductor pj, and note that χ(1 + pj−1) ̸= 1, while
+χ(1 + pj−1)p = χ(1 + pj) = 1, so χ(1 + pj−1) is a nontrivial pth root of
+unity. Since p is odd, χ(1+pj−1) is not a 4th root of unity, so χ cannot
+be quartic and primitive.
+By the above classification, a primitive totally quartic character χ of
+odd conductor must factor over distinct primes pi ≡ 1 (mod 4), and
+the p-part of χ must be χπ or χπ, where ππ = p. We may assume that
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+11
+π and π are primary primes. Hence χ factors as �
+i χπi. The property
+that β := π1 . . . πk is squarefree is equivalent to the condition that the
+πi are distinct. Moreover, the property (β, β) = 1 is equivalent to that
+πiπi = pi ≡ 1 (mod 4), for all i. Hence, every quartic character of odd
+conductor arises uniquely in the form (3.1).
+Next we treat p = 2. There are four primitive quartic characters of
+conductor 24, since (Z/(24))× ≃ Z/(2) × Z/(4). We claim there are no
+primitive quartic characters of conductor 2j, with j ̸= 4. For j ≤ 3 or
+j = 5 this is a simple finite computation. For j ≥ 6, one can show this
+as follows. First, χ(1 + 2j−1) = −1, since χ2(1 + 2j−1) = χ(1 + 2j) = 1,
+and primitivity shows χ(1+2j−1) ̸= 1. By a similar idea, χ(1+2j−2)2 =
+χ(1 + 2j−1) = −1, so χ(1 + 2j−2) = ±i. We finish the claim by noting
+χ2(1 + 2j−3) = χ(1 + 2j−2) = ±i, so χ(1 + 2j−3) is a square-root of
+±i, and hence χ is not quartic. With the claim established, we easily
+obtain the final sentence of the lemma.
+□
+Example 3.3. The first totally quartic primitive character of compos-
+ite conductor has conductor 65. While there are 8 quartic primitive
+characters of conductor 65, the LMFDB labels of the totally quartic
+ones are 65.18, 65.47, 65.8, and 65.57.
+3.2.2. Sextic characters. The construction of sextic characters uses the
+arithmetic in the Eisenstein integers Z[ω], where ω = e2πi/3. The ring
+Z[ω] has class number 1, unit group {±1, ±ω, ±ω2}, and discriminant
+−3. We say α ∈ Z[ω] with (α, 3) = 1 is primary1 if α ≡ 1 (mod 3).
+Warning: our usage of primary is consistent with [HBP79], but conflicts
+with the definition of [IR90]. However, it is easy to translate since α is
+primary in our sense if and only if −α is primary in the sense of [IR90].
+Any element in Z[ω] coprime to 3 has a unique primary associate, which
+comes from the fact that the unit group in the ring Z[ω]/(3) may be
+identified with {±1, ±ω, ±ω2}. An unramified prime p ∈ Z splits as
+p = ππ if and only if p ≡ 1 (mod 3). Given π with N(π) = p, define
+the cubic residue symbol ( α
+π)3 for α ∈ Z[ω] by ( α
+π)3 ∈ {1, ω, ω2} and
+( α
+π)3 ≡ α
+p−1
+3
+(mod π). The map χπ(α) = ( α
+π)3 from (Z[ω]/(π))× to
+{1, ω, ω2} is a character of order 3. The restriction of χπ to Z induces a
+primitive cubic Dirichlet character of conductor p. Note that χπ = χ−π.
+Motivated by the fact that a sextic character factors as a cubic times
+a quadratic, we next discuss the classification of cubic characters.
+1We remark that the usage of primary is context-dependent, and that since we
+do not mix quartic and sextic characters, we hope there will not be any ambiguity
+
+12
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Lemma 3.4. Every primitive cubic Dirichlet character of conductor
+coprime to 3 is of the form χβ, where β = π1 . . . πk is a product of
+distinct primary primes, (β, 3β) = 1, and where
+(3.2)
+χβ(α) =
+�α
+β
+�
+3 =
+k
+�
+i=1
+� α
+πi
+�
+3.
+The cubic primitive characters of conductor divisible by 3 are of the
+form χ3χβ where χ3 is one of two cubic characters of conductor 32,
+and χβ is cubic of conductor coprime to 3.
+Proof. The classification of such characters with conductor coprime to
+3 is given by [BY10, Lemma 2.1], so it only remains to treat cubic
+characters of conductor 3j. The primitive character of conductor 3 is
+not cubic. Next, the group (Z/(9))× is cyclic of order 6, generated by
+2. There are two cubic characters, determined by χ(2) = ω±1. Next
+we argue that there is no primitive cubic character of conductor 3j
+with j ≥ 3. For this, we first observe that χ(1 + 3j−1) = ω±1, since
+primitivity implies χ(1 + 3j−1) ̸= 1, and χ(1 + 3j−1)3 = χ(1 + 3j) = 1.
+Next we have χ(1 + 3j−2)3 = χ(1 + 3j−1) = ω±1, so χ(1 + 3j−2) is a
+cube-root of ω±1. Therefore, χ cannot be cubic.
+□
+3.3. Counting characters. To start, we count all the quartic and sex-
+tic characters of conductor up to some bound and in each family. Such
+counts were found for arbitrary order in [FMS10] by Finch, Martin and
+Sebah, but since we are interested in only quartic and sextic charac-
+ters, in which case the proofs simplify, we prove the results we need.
+Moreover, we need other variants for which we cannot simply quote
+[FMS10], so we will develop a bit of machinery that will be helpful for
+these other questions as well.
+We begin with a lemma based on the Perron formula.
+Lemma 3.5. Suppose that a(n) is a multiplicative function such that
+|a(n)| ≤ dk(n), the k-fold divisor function, for some k ≥ 0. Let Z(s) =
+�
+n≥1 a(n)n−s, for Re(s) > 1. Suppose that for some integer j ≥ 0,
+(s−1)jZ(s) has a analytic continuation to a region of the form {σ+it :
+σ > 1 −
+c
+log(2+|t|)}, for some c > 0. In addition, suppose that Z(s) is
+bounded polynomially in log (2 + |t|) in this region. Then
+(3.3)
+�
+n≤X
+a(n) = XPj−1(log X) + O(X(log X)−100),
+for Pj−1 some polynomial of degree ≤ j − 1 (interpreted as 0, if j = 0).
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+13
+The basic idea is standard, yet we were unable to find a suitable refer-
+ence.
+Proof sketch. One begins by a use of the quantitative Perron formula,
+for which a convenient reference is [MV07, Thm. 5.2]. This implies
+(3.4)
+�
+n≤X
+a(n) =
+1
+2πi
+� σ0+iT
+σ0−iT
+Z(s)Xsds
+s + R,
+where R is a remainder term, and we take σ0 = 1+
+c
+log X . Using [MV07,
+Cor. 5.3] and standard bounds on mean values of dk(n), one can show
+R ≪ X
+T Poly(log X). Next one shifts the contour of integration to the
+line 1 − c/2
+log T . The pole (if it exists) of Z(s) leads to a main term of the
+form XPj−1(log X), as desired. The new line of integration is bounded
+by
+(3.5)
+Poly(log T)X1− c/2
+log T .
+Choosing log T = (log X)1/2 gives an acceptable error term.
+□
+3.3.1. Quartic characters. Let Ψtot,odd
+4
+(X) ⊆ Ψtot
+4 (X) denote the sub-
+set of characters with odd conductor.
+Proposition 3.6. For some constants Ktot
+4 , Ktot,odd
+4
+> 0, we have
+(3.6)
+|Ψtot
+4 (X)| ∼ Ktot
+4 X,
+and
+|Ψtot,odd
+4
+(X)| ∼ Ktot,odd
+4
+X.
+Moreover,
+(3.7)
+|Ψ′
+4(X)| ∼
+X
+log X .
+Proof. By Lemma 3.2,
+(3.8)
+|Ψtot,odd
+4
+(X)| =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+N(β)≤X
+1,
+and
+(3.9)
+|Ψtot
+4 (X)| = |Ψtot,odd
+4
+(X)| + 4|Ψtot,odd
+4
+(2−4X)|.
+
+14
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+To show (3.6), it suffices to prove the asymptotic formula for |Ψtot,odd
+4
+(X)|.
+In view of Lemma 3.5, it will suffice to understand the Dirichlet series
+(3.10)
+Z4(s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+1
+N(β)s =
+�
+π̸=π
+(π,2)=1
+(1 + N(π)−s) =
+�
+p≡1 (mod 4)
+(1 + p−s)2.
+Let χ4 be the primitive character modulo 4, so that ζ(s)L(s, χ4) =
+ζQ[i](s). Then
+(3.11) Z4(s) = ζQ[i](s)
+�
+p
+(1 − p−s)(1 − χ4(p)p−s)
+�
+p≡1 (mod 4)
+(1 + p−s)2,
+which can be simplified as
+(3.12)
+Z4(s) = ζQ[i](s)ζ−1(2s)(1 + 2−s)−1
+�
+p≡1 (mod 4)
+(1 − p−2s).
+Therefore, Z4(s) has a simple pole at s = 1, and its residue is a positive
+constant. Moreover, the standard analytic properties of ζQ[i](s) let us
+apply Lemma 3.5, giving the result.
+The asymptotic on Ψ′
+4(X) follows from the prime number theorem in
+arithmetic progressions, since there are two quartic characters of prime
+conductor p ≡ 1 (mod 4), and none with p ≡ 3 (mod 4).
+□
+Lemma 3.7. We have
+(3.13)
+|Ψ4(X)| = K4X log X + O(X),
+for some K4 > 0
+Proof. Every primitive quartic character factors uniquely as χ4χ2 with
+χ4 totally quartic of conductor q4 > 1 and χ2 quadratic of conductor
+q2, with (q4, q2) = 1. It is convenient to drop the condition q4 > 1,
+thereby including the quadratic characters; this is allowable since the
+number of quadratic characters is O(X), which is acceptable for the
+claimed error term.
+The Dirichlet series for |Ψ4(X)|, modified to include the quadratic char-
+acters, is
+(3.14)
+Zall
+4 (s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+1
+N(β)s
+�
+q2∈Z≥1
+(q2,2N(β))=1
+1
+qs
+2
+.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+15
+A calculation with Euler products shows Zall
+4 (s) = ζQ[i](s)ζ(s)A(s),
+where A(s) is given by an absolutely convergent Euler product for
+Re(s) > 1/2. Since Zall
+4 (s) has a double pole at s = 1, this shows the
+claim, using Lemma 3.5.
+□
+3.3.2. Sextic characters. Next we turn to the sextic case. The proof of
+the following proposition is similar to the proof of Proposition 3.6 and
+so we omit it here.
+Proposition 3.8. For some Ktot
+6
+> 0, we have
+(3.15)
+|Ψtot
+6 (X)| ∼ Ktot
+6 X,
+and
+|Ψ′
+6(X)| ∼
+X
+log X .
+A primitive totally sextic character factors uniquely as a primitive cubic
+character (with odd conductor, since 2 ̸≡ 1 (mod 3)), times the Jacobi
+symbol of the same modulus as the cubic character.
+In general, a
+primitive sextic character factors uniquely as χ6χ3χ2 of modulus q6q3q2,
+pairwise coprime, with χ6 totally sextic of conductor q6, χ3 cubic of
+conductor q3, and χ2 quadratic of conductor q2.
+Lemma 3.9. We have |Ψ6(X)| = K6X(log X)2+O(X log X), for some
+K6 > 0.
+Proof. Write χ = χ6χ3χ2 as above. Note that membership in Ψ6(X)
+requires q6 > 1, which is an unpleasant condition when working with
+Euler products. However, the number of χ = χ3χ2, i.e., with χ6 = 1,
+is O(X log X), so we may drop the condition q6 > 1 when estimating
+|Ψ6(X)|.
+For simplicity, we count the characters with q2 odd and (q6q3, 3) =
+1; the general case follows similar lines. The Dirichlet series for this
+counting function is
+Zall
+6 (s) =
+�
+0̸=(β6)⊆Z[ω]
+(β6,3β6)=1
+β6 squarefree
+1
+N(β6)s
+�
+0̸=(β3)⊆Z[ω]
+(β3,3β3)=1
+β3 squarefree
+(N(β3),N(β6)=1
+1
+N(β3)s
+�
+q2∈Z≥1
+(q2,2N(β3β6))=1
+1
+qs
+2
+.
+A calculation with Euler products shows Zall
+6 (s) = ζQ[ω](s)2ζ(s)A(s),
+where A(s) is given by an absolutely convergent Euler product for
+Re(s) > 1/2. Since Zall
+6 (s) has a triple pole at s = 1, this shows the
+claim, using Lemma 3.5.
+□
+
+16
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+3.4. Equidistribution of Gauss sums. We first focus on the quartic
+case, and then turn to the sextic case.
+3.4.1. Quartic characters. The following standard formula can be found
+as [IK04, (3.16)].
+Lemma 3.10. Suppose that χ = χ1χ2 has conductor q = q1q2, with
+(q1, q2) = 1, and χi of conductor qi. Then
+(3.16)
+τ(χ1χ2) = χ2(q1)χ1(q2)τ(χ1)τ(χ2).
+Corollary 3.11. Let notation be as in Lemma 3.10. Suppose that χ is
+totally quartic and q is odd. Then
+(3.17)
+τ(χ1χ2)2 = τ(χ1)2τ(χ2)2.
+Proof. By Lemma 3.10, we will obtain the formula provided χ2
+2(q1)χ2
+1(q2) =
+1. Note that χ2
+i is the Jacobi symbol, so χ2
+2(q1)χ2
+1(q2) = ( q1
+q2)( q2
+q1) = 1,
+by quadratic reciprocity, using that q1 ≡ q2 ≡ 1 (mod 4).
+□
+Lemma 3.12. Suppose π ∈ Z[i] is a primary prime, with N(π) = p ≡
+1 (mod 4). Let χπ(x) = [ x
+π] be the quartic residue symbol. Then
+(3.18)
+τ(χπ)2 = −χπ(−1)√pπ.
+More generally, if β is primary, squarefree, with (β, 2β) = 1, then
+(3.19)
+τ(χβ)2 = µ(β)χβ(−1)
+�
+N(β)β.
+Proof. The formula for χπ follows from [IR90, Thm.1 (Chapter 8),
+Prop. 9.9.4]. The formula for general β follows from Corollary 3.11
+and Lemma 3.2.
+□
+Lemma 3.13. Suppose that χ = χ2χ4 is a primitive quartic character
+with odd conductor q, with χ2 quadratic of conductor q2, χ4 totally
+quartic of conductor q4, and with q2q4 = q.
+Then
+(3.20)
+τ(χ)2 =
+�−q4
+q2
+�
+q2τ(χβ)2.
+Proof. By Lemma 3.10, we have τ(χ)2 = χ2(q4)2χ4(q2)2τ(χ2)2τ(χ4)2.
+To simplify this, note χ2(q4)2 = 1, χ2
+4(q2) = (q2
+q4) = (q4
+q2), and τ(χ2)2 =
+ϵ2
+q2q2 = ( −1
+q2 )q2.
+□
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+17
+Our next goal is to express τ(χβ)2 in terms of a Hecke Grossencharacter.
+Define
+(3.21)
+λ∞(α) = α
+|α|,
+α ∈ Z[i], α ̸= 0.
+Next define a particular character λ1+i : R× → S1, where R = Z[i]/(1+
+i)3, by
+(3.22)
+λ1+i(ik) = i−k,
+k ∈ {0, 1, 2, 3}.
+This indeed defines a character since R× ≃ Z/4Z, generated by i. For
+α ∈ Z[i], (α, 1 + i) = 1, define
+(3.23)
+λ((α)) = λ1+i(α)λ∞(α).
+For this to be well-defined, we need that the right hand side of (3.23)
+is constant on units in Z[i]. This is easily seen, since λ∞(ik) = ik =
+λ1+i(ik)−1. Therefore, λ defines a Hecke Grossencharacter, as in [IK04,
+Section 3.8]. Moreover, we note that
+(3.24)
+τ(χβ)2
+N(β) = µ(β)
+�
+2
+N(β)
+�
+λ((β))
+since this agrees with (3.19) for β primary, and is constant on units.
+According to [IK04, Theorem 3.8], the Dirichlet series
+(3.25)
+L(s, λk) =
+�
+0̸=(β)⊆Z[i]
+λ((β))k
+N(β)s ,
+(k ∈ Z),
+defines an L-function having analytic continuation to s ∈ C with no
+poles except for k = 0. The same statement holds when twisting λk by
+a finite-order character.
+For k ∈ Z, define the Dirichlet series
+(3.26)
+Z(k, s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+(τ(χβ)2/N(β))k
+N(β)s
+,
+Re(s) > 1.
+Proposition 3.14. Let δk = −1 for k odd, and δk = +1 for k even.
+We have
+(3.27) Z(k, s) = A(k, s)L(s, (λ · χ2)k)δk,
+where
+χ2(β) =
+�
+2
+N(β)
+�
+,
+and where A(k, s) is given by an Euler product absolutely convergent
+for Re(s) > 1/2.
+
+18
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+In particular, the zero free region (as in [IK04, Theorem 5.35]) implies
+that Z(k, s) is analytic in a region of the type postulated in Lemma
+3.5. Moreover, the proof of [MV07, Theorem 11.4] shows that Z(k, s)
+is bounded polynomially in log(2 + |t|) in this region.
+Proof. The formula (3.24) shows that Z(k, s) has an Euler product of
+the form
+(3.28)
+Z(k, s) =
+�
+(π)̸=(π)
+(1 + (−1)k χk
+2(π)λk((π))
+N(π)s
+).
+This is an Euler product over the split primes in Z[i]. We extend this
+to include the primes p ≡ 3 (mod 4) as well, with N(π) = p2. It is
+convenient to define χ2(1 + i) = 0, so we can freely extend the product
+to include the ramified prime 1 + i. In all, we get
+(3.29)
+Z(k, s) =
+� �
+p
+(1 − χk
+2(p)λk(p)
+N(p)s
+)
+�−δk �
+p
+(1 + O(p−2s)).
+Note the product over p is L(s, (λ · χ2)k)δk, as claimed.
+□
+According to Weyl’s equidistribution criterion [IK04, Ch. 21.1], a se-
+quence of real numbers θn, 1 ≤ n ≤ N is equidistributed modulo 1 if
+and only if �
+n≤N e(kθn) = o(N) for each integer k ̸= 0. We apply
+this to e(θn) = (τ(χ)2/q), whence e(kθn) = (τ(χ)2/q)k. Due to the
+twisted multiplicativity formula (3.16), the congruence class in which
+2k lies modulo ℓ may have a simplifying effect on τ(χ)2k. For instance,
+when ℓ = 4, then k even leads to a simpler formula than k odd. This
+motivates treating these cases separately. As a minor simplification,
+below we focus on the sub-family of characters of odd conductor. The
+even conductor case is only a bit different.
+Corollary 3.15. The Gauss sums τ(χ)2/q for χ totally quartic of odd
+conductor q, equidistribute on the unit circle.
+Proof. The complex numbers τ(χ)2/q lie on the unit circle.
+Weyl’s
+equidistribution criterion says that these normalized squared Gauss
+sums equidistribute on the unit circle provided
+(3.30)
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+N(β)≤X
+(τ(χβ)2/N(β))k = o(X),
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+19
+Figure 2. This histogram represents the distribution
+of the argument of the τ(χ)2/cond(χ) for totally quartic
+characters. Each histogram is made by calculating the
+Gauss sums of characters of each order up to prime and
+composite conductor 300000.
+for each nonzero integer k. In turn, this bound is implied by Propo-
+sition 3.14, using the zero-free region for the Hecke Grossencharacter
+L-functions in [IK04, Theorem 5.35].
+□
+To contrast this, we will show that the normalized Gauss sums τ(χ)2/q,
+with χ ranging over all quartic characters, equidistribute slowly. More
+precisely, we have the following result.
+Proposition 3.16. Let k ∈ 2Z, k ̸= 0. There exists ck ∈ C such that
+(3.31)
+�
+q≤X
+(q,2)=1
+�
+χ:χ4=1
+cond(χ)=q
+(τ(χ)2/q)k = ckX + o(X).
+Remark 3.17. Recall from Lemma 3.7 that the total number of such
+characters grows like X log X, so Proposition 3.16 shows that the rate
+of equidistribution is only O((log X)−1) here. In contrast, in the family
+of totally quartic characters, the GRH would imply a rate of equidis-
+tribution of the form O(X−1/2+ε). This difference in rates of equidis-
+tribution is supported by Figure 2 in which we see that the arguments
+
+5+00
+4000
+DODE
+DOZ
+1000
+E-
+-1
+020
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+of squares of the Gauss sums of totally quartic characters quickly con-
+verge to being uniformly distributed, as compared to the Gauss sums
+of all quartic characters.
+In addition, one can derive a similar result when restricting to χ ∈
+Ψ4(X), simply by subtracting off the contribution from the quadratic
+characters alone.
+Proof. As in Lemma 3.13, write χ = χ2χ4, with χ2 quadratic and χ4
+totally quartic. Then τ(χ)4/(q1q2)2 = τ(χ4)4/q2
+2. The analog of Z(k, s),
+using k even to simplify, is
+(3.32)
+Zall(k, s) =
+�
+0̸=(β)⊆Z[i]
+(β,2β)=1
+β squarefree
+τ(χβ)2k/N(β)k
+N(β)s
+�
+q2∈Z≥1
+(q2,2N(β))=1
+1
+qs
+2
+.
+Referring to the calculation in Proposition 3.14, we obtain
+(3.33)
+Zall(k, s) = ζ(s)L(s, λk)A(s),
+where A(s) is an Euler product absolutely convergent for Re(s) > 1/2.
+Since this generating function has a simple pole at s = 1, we deduce
+Proposition 3.16.
+□
+As mentioned above, in order to deduce equidistribution, by Weyl’s
+equidistribution criterion, we also need to consider odd values of k in
+(3.31). This is more technical than the case for even k, so we content
+ourselves with a conjecture.
+Conjecture 3.18. For each odd k, there exists δ > 0 such that
+(3.34)
+�
+q≤X
+(q,2)=1
+�
+χ:χ4=1
+cond(χ)=q
+(τ(χ)2/q)k ≪k,δ X1−δ.
+Remark 3.19. By Lemma 3.13 and (3.24), this problem reduces to
+understanding sums of the rough shape
+�
+β,q2
+q2N(β)≤X
+��−N(β)
+q2
+�
+µ(β)
+�
+2
+N(β)
+�
+λ((β))k,
+where we have omitted many of the conditions on β and q2. In the
+range where q2 is very small, the GRH gives cancellation in the sum
+over β. Conversely, in the range where N(β) is very small, the GRH
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+21
+gives cancellation in the sum over q2. This discussion indicates that
+Conjecture 3.18 follows from GRH, with any δ < 1/4.
+Unconditionally, one can deduce some cancellation using the zero-free
+region for the β-sum (with q2 very small), and a subconvexity bound
+for the q2-sum (with N(β) very small). In the range where both q2
+and N(β) have some size, then Heath-Brown’s quadratic large sieve
+[HB95] gives some cancellation.
+Since we logically do not need an
+unconditional proof of equidistribution, we omit the details for brevity.
+Remark 3.20. Conjecture 3.18 and Proposition 3.16 together imply
+that the squares of the quartic Gauss sums do equidistribute in the full
+family Ψ4(X).
+3.4.2. Sextic characters. Now we turn to the sextic Gauss sums.
+Lemma 3.21. Suppose that χ is totally sextic of conductor q, and say
+χ = χ2χ3 with χ2 quadratic and χ3 cubic, each of conductor q. Suppose
+χ3 = χβ, as in Lemma 3.4. Then
+(3.35)
+τ(χ) = µ(q)χ3(2)τ(χ2)τ(χ3)βq−1.
+Proof. By [IK04, (3.18)], τ(χ2)τ(χ3) = J(χ2, χ3)τ(χ), where J(χ2, χ3)
+is the Jacobi sum.
+It is easy to show using the Chinese remainder
+theorem that if χ2 = �
+p χ(p)
+2
+and χ3 = �
+p χ(p)
+3 , then
+(3.36)
+J(χ2, χ3) =
+�
+p
+J(χ(p)
+2 , χ(p)
+3 ).
+The Jacobi sum for characters of prime conductor can be evaluated
+explicitly using the following facts. By [Lem00, Prop. 4.30],
+(3.37)
+J(χ(p)
+2 , χ(p)
+3 ) = χ(p)
+3 (22)J(χ(p)
+3 , χ(p)
+3 ).
+Suppose that χ(p)
+3
+= χπ, where ππ = p, and π is primary. Then [IR90,
+Ch. 9, Lem. 1] implies J(χπ, χπ) = −π. (Warning: they state the
+value π instead of −π, but recall their definition of primary is opposite
+our convention. Also recall that χπ = χ−π.) Gathering the formulas,
+we obtain
+(3.38)
+τ(χ2)τ(χ3) = τ(χ)χ3(2)2 �
+πi|β
+(−πi) = τ(χ)χ3(2)2µ(q)β.
+Rearranging this and using ββ = q completes the proof.
+□
+
+22
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Corollary 3.22. Let conditions be as in Lemma 3.21. Then
+(3.39)
+τ(χ)2/q = χ3(4)
+�−1
+q
+�
+τ(χβ)2β
+2/q2.
+Patterson [Pat78] showed that τ(χβ)/√q is uniformly distributed on
+the unit circle, as χβ ranges over primitive cubic characters. The same
+method gives equidistribution after multiplication by a Hecke Grossen-
+character, and so similarly to the quartic case above, we deduce:
+Corollary 3.23 (Patterson). The Gauss sums τ(χ)2/q, for χ totally
+sextic of conductor q, equidistribute on the unit circle.
+In light of Corollary 3.22, Proposition 3.16, and Conjecture 3.18, it
+seems reasonable to conjecture that the points τ(χ)2/q are equidis-
+tributed on the unit circle, as χ varies over all sextic characters. To
+see a limitation in the rate of equidistribution, it is convenient to con-
+sider τ(χ)6/q3, which is multiplicative for χ sextic. For q ≡ 1 (mod 4),
+and χ = χ2 quadratic, we have τ(χ2)2/q = 1, so the quadratic part is
+constant. For χ cubic and q ≡ 1 (mod 4),
+(3.40)
+τ(χβ)6/q3 = µ(β)τ(χβ)3β
+3 = q−1β
+2,
+which is nearly a Hecke Grossencharacter. A similar formula holds for
+χ totally sextic, namely
+(3.41)
+τ(χ)6/q3 = q−4β
+8.
+Therefore, carrying out the same steps as in Proposition 3.16 shows
+that
+(3.42)
+�
+q≤X
+q≡1 (mod 4)
+�
+χ∈Ψ6
+cond(χ)=q
+�
+τ(χ)6/q3�k
+= CkX + o(X).
+This is less of an obstruction than in the quartic case, since here the
+rate of equidistribution is O((log X)−2) instead of O((log X)−1), due to
+the fact that |Ψ6(X)| is approximately log X times as large as |Ψ4(X)|.
+Similarly to the discussion of the quartic case in Remarks 3.19 and
+3.20, we make the following conjecture without further explanation.
+Conjecture 3.24. The Gauss sums τ(χ)2/q, for χ ranging in Ψ6(X),
+equidistribute on the unit circle.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+23
+3.5. Estimates for quartic and sextic characters. In order to ap-
+ply the random matrix theory conjectures, we need variants on Propo-
+sition 3.6, Lemma 3.7, Proposition 3.8, and Lemma 3.9, as follows.
+Lemma 3.25. For primitive Dirichlet characters χ of order ℓ we have
+for ℓ = 4 and ℓ = 6 that
+(3.43)
+�
+χ∈Ψℓ(X)
+1
+�
+cond(χ)
+∼ 2Kℓ
+√
+X(log X)d(ℓ)−2,
+and
+(3.44)
+�
+χ∈Ψtot
+ℓ
+(X)
+1
+�
+cond(χ)
+∼ 2Ktot
+ℓ
+√
+X,
+�
+χ∈Ψ′
+ℓ(X)
+1
+�
+cond(χ)
+∼ 2
+√
+X
+log X .
+Proof. These estimates follow from a straightforward application of
+partial summation or from a minor modification of Lemma 3.5 since
+the generating Dirichlet series for one of these sums has its pole at
+s = 1/2 instead of at s = 1.
+□
+4. Random matrix theory: Conjectural asymptotic
+behavior
+This section closely follows the exposition of §3 of [DFK04] and §4 of
+[DFK07].
+Let U(N) be the set of unitary N×N matrices with complex coefficients
+which forms a probability space with respect to the Haar measure.
+For a family of L-functions with symmetry type U(N), Katz and Sar-
+nak conjectured that the statistics of the low-lying zeros should agree
+with those of the eigenangles of random matrices in U(N) [KS99]. Let
+PA(λ) = det(A − λI) be the characteristic polynomial of A. Keating
+and Snaith [KS00] suggest that the distribution of the values of the L-
+functions at the critical point is related to the value distribution of the
+characteristic polynomials |PA(1)| with respect to the Haar measure on
+U(N).
+For any s ∈ C we consider the moments
+MU(s, N) :=
+�
+U(N)
+|PA(1)|s dHaar
+
+24
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+for the distribution of |PA(1)| in U(N) with respect to the Haar mea-
+sure. In [KS00], Keating and Snaith proved that
+(4.1)
+MU(s, N) =
+N
+�
+j=1
+Γ(j)Γ(j + s)
+Γ2(j + s/2) ,
+so that MU(s, N) is analytic for Re(s) > −1 and has meromorphic
+continuation to the whole complex plane. The probability density of
+|PA(1)| is given by the Mellin transform
+pU(x, N) =
+1
+2πi
+�
+Re(s)=c
+MU(s, N)x−s−1 ds,
+for some c > −1.
+In the applications to the vanishing of twisted L-functions we consider
+in this paper, we are only interested in small values of x where the value
+of pU(x, N) is determined by the first pole of MU(s, N) at s = −1. More
+precisely, for x ≤ N −1/2, one can show that
+pU(x, N) ∼ G2(1/2)N 1/4
+as N → ∞,
+where G(z) is the Barnes G-function with special value [Bar99]
+G(1/2) = exp
+�3
+2ζ′(−1) − 1
+4 log π + 1
+24 log 2
+�
+.
+We will now consider the moments for the special values of twists of
+L-functions.
+We then define, for any s ∈ C, the following sum of
+evaluations at s = 1 of L-functions primitive order ℓ characters of
+conductor less than X:
+(4.2)
+ME(s, X) =
+1
+#FΨℓ,E(X)
+�
+L(E,s,χ)∈FΨℓ,E(X)
+|L(E, 1, χ)|s.
+Then, since the families of twists of order ℓ are expected to have unitary
+symmetry, we have
+Conjecture 4.1 (Keating and Snaith Conjecture for twists of order
+ℓ). With the notation as above,
+ME(s, X) ∼ aE(s/2)MU(s, N)
+as N = 2 log X → ∞,
+where aE(s/2) is an arithmetic factor depending only on the curve E.
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+25
+From Conjecture 4.1, the probability density for the distribution of the
+special values |L(E, 1, χ)| for characters of order ℓ is
+pE(x, X)
+=
+1
+2πi
+�
+Re(s)=c
+ME(s, X)x−s−1 ds
+(4.3)
+∼
+1
+2πi
+�
+Re(s)=c
+aE(s/2)MU(s, N)x−s−1 ds
+(4.4)
+as N = 2 log X → ∞. As above, when x ≤ N −1/2, the value of pE(x, X)
+is determined by the residue of MU(s, N) at s = −1, thus it follows
+from (4.4) that for x ≤ (2 log X)−1/2,
+(4.5)
+pE(x, X) ∼ 21/4aE(−1/2)G2(1/2) log1/4(X)
+as X → ∞.
+We now use the probability density of the random matrix model with
+the properties of the integers nE(χ) to obtain conjectures for the van-
+ishing of the L-values |L(E, 1, χ)|. When χ is either quartic or sextic,
+the discretization nE(χ) is a rational integer since Z[ζℓ] ∩ R = Z when
+ℓ = 4 or 6.
+Lemma 4.2. Let χ be a primitive Dirichlet character of order ℓ = 4
+or 6. Then
+|L(E, 1, χ)| =
+cE,ℓ
+�
+cond(χ)
+|nE(χ)|,
+where cE,ℓ is a nonzero constant which depends only on the curve E
+and ℓ.
+Proof. By rearranging equation (2.2) we obtain
+|L(E, 1, χ)| =
+����
+Ωϵ(E) τ(χ) kE nE(χ)
+cond(χ)
+���� = |Ωϵ(E) kE nE(χ)|
+�
+cond(χ)
+= cE,ℓ|nE(χ)|
+�
+cond(χ)
+,
+where the nonzero constant kE is that of Proposition 2.2.
+□
+We write
+(4.6)
+Prob{|L(E, 1, χ)| = 0} = Prob{|L(E, 1, χ)| < B(cond(χ))},
+for some function B(cond(χ)) of the character. By Lemma 4.2 we may
+take B(cond(χ)) =
+cE,ℓ
+�
+cond(χ)
+. Note that since cE,ℓ ̸= 0, if
+|nE(χ)|cE,ℓ
+�
+cond(χ)
+<
+cE,ℓ
+�
+cond(χ)
+,
+then |nE(χ)| < 1 and hence must vanish since |nE(χ)| ∈ Z≥0.
+
+26
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+Using (4.5), we have
+Prob{|L(E, 1, χ)| = 0}
+=
+� B(cond(χ))
+0
+21/4aE(−1/2)G2(1/2) log1/4(X) dx
+=
+21/4aE(−1/2)G2(1/2) log1/4(X)B(cond(χ))
+Summing the probabilities gives
+|VΨℓ,E(X)| = 21/4cE,kaE(−1/2)G2(1/2) log1/4(X)
+�
+cond(χ)≤X
+1
+�
+cond(χ)
+.
+Thus, by the analysis in §3.3, we have
+|VΨ4,E(X)| ∼ 25/4cE,4K4aE(−1/2)G2(1/2) log1/4(X)
+√
+X log X
+∼ bE,4X1/2 log5/4 X
+and
+|VΨ6,E(X)| ∼ 25/4cE,6K6aE(−1/2)G2(1/2) log1/4(X)
+√
+X(log X)2
+∼ bE,6X1/2 log9/4 X
+as X → ∞.
+Moreover, if we restrict to those characters that are totally quartic or
+sextic, we get the following estimates
+|VΨtot
+4 ,E(X)| ∼ 25/4cE,4Ktot
+4 aE(−1/2)G2(1/2) log1/4(X)
+√
+X
+∼ btot
+E,4X1/2 log1/4 X
+and
+|VΨtot
+6 ,E(X)| ∼ 25/4cE,6Ktot
+6 aE(−1/2)G2(1/2)
+∼ btot
+E,6X1/2 log1/4 X
+as X → ∞.
+Finally, if we restrict only to those twists by characters of prime con-
+ductor, we conclude
+|VΨ′
+4,E(X)| ∼ 25/4cE,4aE(−1/2)G2(1/2) log1/4(X)
+√
+X
+log X
+∼ b′
+E,4X1/2 log−3/4 X
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+27
+and
+|VΨ′
+6,E(X)| ∼ 25/4cE,6aE(−1/2)G2(1/2) log1/4(X)
+√
+X
+log X
+∼ b′
+E,6X1/2 log−3/4 X
+as X → ∞.
+4.1. Computations. Here we provide numerical evidence for Conjec-
+ture 1.1. The computations of the Conrey labels for the characters were
+done in SageMath [Sag21] and the computations of the L-functions were
+done in PARI/GP [PAR22]. The L-function computations were done
+in a distributed way on the Open Science Grid. For each curve, we
+generated a PARI/GP script to calculate a twisted L-function for each
+primitive character of order 4 and 6, and then combined the results into
+one file at the end. The combined wall time of all the computations
+was more than 50 years. The code and data are available at [BR23].
+In Figure 3 we plot the points
+(X, X1/2 log5/4 X
+|VΨ4,11.a.1(X)|), (X, X1/2 log−3/4 X
+|VΨ′
+4,11.a.1(X)| ), (X,
+X1/2 log1/4 X
+|VΨtot
+4
+,11.a.1(X)|)
+that provides a comparison between the predicted vanishings of L(E, 1, χ)
+for quartic characters and for the curve 11.a.1. In Figure 4 we plot the
+analogous points for the same curve but for sextic twists. In Figure 5
+we plot the points
+(X, X1/2 log−3/4 X
+|VΨ′
+4,37.a.1(X)| ), (X, X1/2 log−3/4 X
+|VΨ′
+6,37.a.1(X)| )
+Even though we are most interested in the families of all quartic and
+sextic twists, we include the families of twists of prime conductor be-
+cause there are far fewer such characters and so we can calculate the
+number of vanishings up to a much larger X. We include the fami-
+lies of twists by totally quartic and sextic characters to highlight the
+transition between the family of prime conductors and the family of all
+conductors.
+References
+[Bar99]
+E.W. Barnes. The theory of the G-function. Quart. J. Math., 31:264–314,
+1899.
+[BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.
+On the modularity of elliptic curves over Q: wild 3-adic exercises. Journal
+of the American Mathematical Society, pages 843–939, 2001.
+[BE81]
+Bruce C Berndt and Ronald J Evans. The determination of Gauss sums.
+Bulletin of the American Mathematical Society, 5(2):107–129, 1981.
+
+28
+JENNIFER BERG, NATHAN C. RYAN, AND MATTHEW P. YOUNG
+(a)
+The
+ratio
+of
+predicted
+vanishings
+to
+empirical
+van-
+ishings
+of
+twists
+of
+the curve 11.a.1 by
+quartic characters of
+conductor ≤ 700000.
+(b) The ratio of pre-
+dicted
+vanishings
+to
+empirical
+vanishings
+of twists of the curve
+11.a.1
+by
+quartic
+characters
+of
+prime
+conductor ≤ 2000000.
+(c) The ratio of pre-
+dicted
+vanishings
+to
+empirical
+vanishings
+of twists of the curve
+11.a.1
+by
+totally
+quartic characters of
+conductor ≤ 700000.
+Figure 3. Verification of Conjecture 1.1 for quartic
+twists of 11.a.1.
+(a) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+11.a.1 by sextic char-
+acters of conductor ≤
+300000.
+(b) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+11.a.1 by sextic char-
+acters of prime con-
+ductor ≤ 2000000.
+(c) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+11.a.1 by totally sex-
+tic characters of con-
+ductor ≤ 300000.
+Figure 4. Verification of Conjecture 1.1 for sextic
+twists of 11.a.1.
+[BR23]
+Jen Berg and Nathan C. Ryan. Code and data for quartic and sextic
+twists of elliptic curve L-functions. http://eg.bucknell.edu/~ncr006/
+quartic-sextic-twists-website/, 2023.
+[BY10]
+Stephan Baier and Matthew P. Young. Mean values with cubic characters.
+J. Number Theory, 130(4):879–903, 2010.
+[CFK+05] J. Brian Conrey, David W Farmer, Jon P Keating, Michael O Rubin-
+stein, and Nina C Snaith. Integral moments of L-functions. Proceedings
+of the London Mathematical Society, 91(1):33–104, 2005.
+[Cho87]
+Sarvadaman Chowla. The Riemann hypothesis and Hilbert’s tenth prob-
+lem, volume 4. CRC Press, 1987.
+
+0.38
+0.36 
+tE'O
+0.32
+0.30
+0.28 
+0.26
+0.D0
+0.25
+0.50
+0.75
+LDo
+125
+150
+175
+20o
+1e614
+12
+LD
+0.B 
+0.6 
+0
+ 11
+9 -
+8 -
+1 
+61
+50000
+150000
+DO
+250000
+3000000.45
+0.40 -
+0.35 
+0.30 
+0.25
+0.D0
+0.25
+0.50
+0.75
+Lio
+125
+150
+175
+200
+1e616
+15
+14
+13
+12
+11
+LD
+0
+DODS
+10dC0
+150000
+240000
+25000
+3+0dC04.5
+4.D
+3.5-
+3.0
+25 -
+20
+15
+LD
+   VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+29
+(a) The ratio of pre-
+dicted
+vanishings
+to
+empirical
+vanishings
+of twists of the curve
+37.a.1
+by
+quartic
+characters
+of
+prime
+conductor ≤ 2000000.
+(b) The ratio of pre-
+dicted
+vanishings
+to
+empirical vanishings of
+twists
+of
+the
+curve
+37.a.1 by sextic char-
+acters of prime con-
+ductor ≤ 2000000.
+Figure 5. Verification of parts of Conjecture 1.1 for
+twists of 37.a.1.
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+0.DO
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+LDo
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+matics. Springer-Verlag, Berlin, 2000. From Euler to Eisenstein.
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+from http://pari.math.u-bordeaux.fr/.
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+Samuel J Patterson. The distribution of general Gauss sums and simi-
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+Sage Developers. SageMath, the Sage Mathematics Software System (Ver-
+sion 9.4), 2021. https://www.sagemath.org.
+[SBH+09] Igor Sfiligoi, Daniel C Bradley, Burt Holzman, Parag Mhashilkar, San-
+jay Padhi, and Frank Wurthwein. The pilot way to grid resources using
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+Email address: jsb047@bucknell.edu
+Email address: nathan.ryan@bucknell.edu
+Department of Mathematics, Bucknell University, Lewisburg, PA 17837
+Email address: myoung@math.tamu.edu
+
+VANISHING OF QUARTIC AND SEXTIC TWISTS OF L-FUNCTIONS
+31
+Department of Mathematics, Texas A&M University, College Station,
+TX 77843-3368
+

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+Practical challenges in data-driven
+interpolation: dealing with noise, enforcing
+stability, and computing realizations
+Quirin Aumann∗
+Ion Victor Gosea†
+∗Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg,
+Germany.
+Email: aumann@mpi-magdeburg.mpg.de, ORCID: 0000-0001-7942-5703
+†Max Planck Instiute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg,
+Germany.
+Email: gosea@mpi-magdeburg.mpg.de, ORCID: 0000-0003-3580-4116
+Abstract: In this contribution, we propose a detailed study of interpolation-based data-
+driven methods that are of relevance in the model reduction and also in the systems and
+control communities. The data are given by samples of the transfer function of the under-
+lying (unknown) model, i.e., we analyze frequency-response data. We also propose novel
+approaches that combine some of the main attributes of the established methods, for ad-
+dressing particular issues.
+This includes placing poles and hence, enforcing stability of
+reduced-order models, robustness to noisy or perturbed data, and switching from different
+rational function representations. We mention here the classical state-space format and
+also various barycentric representations of the fitted rational interpolants. We show that
+the newly-developed approaches yield, in some cases, superior numerical results, when com-
+paring to the established methods. The numerical results include a thorough analysis of
+various aspects related to approximation errors, choice of interpolation points, or placing
+dominant poles, which are tested on some benchmark models and data-sets.
+Keywords: Data-driven methods, rational approximation, interpolatory methods, least
+squares fit, Loewner framework, frequency response data, pole placement, noisy measure-
+ments, Loewner and Cauchy matrices.
+Novelty statement: This note shows that by combining the features of established data-
+driven rational approximation methods based on interpolation (and/or least squares fit),
+one can devise methods that offer additional important advantages. These include stabil-
+ity enforcement by placing poles in an elegant and numerically stable manner, together
+with robustness to noisy data.
+1. Introduction
+Approximation of large-scale dynamical systems is pivotal for serving the scopes of efficient simulation
+and designing control laws in real-time. The technique for reducing the complexity of a system is known
+as model order reduction (MOR) [1, 5, 13, 14]. There exist a number of methodologies for reducing
+large-scale models, and each method is tailored to some specific applications (mostly, but not restricted
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+arXiv:2301.04906v1  [math.NA]  12 Jan 2023
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+2
+to mechanical and electrical engineering) and to achieving certain goals (stability, passivity or structure
+preservation), on top of the complexity reduction part. Data-driven MOR approaches are of particular
+importance when access to high-fidelity models is not explicitly granted. This means that a state-
+space formulation with access to internal variables is not available, yet input/output data are. Such
+methods circumvent the need to access an exact description of the original model and are applicable
+whenever classical projection-based MOR is not. Here, we mention system and control methodologies
+that are based on interpolation or least-square fit of data (i.e., frequency response measurements),
+such as vector fitting [31], the Loewner framework [41], or the AAA algorithm [42]. Methods that use
+time-domain data are also of interest, including the ones that require input-output data together with
+those which use snapshot data (access to the state evolution), such as the classical ones in [36,58,59],
+followed by [8], [47] or [54].
+We focus on interpolation-based or so-called moment matching (MM) methods which have emerged,
+were developed, and improved continuously in the last decades. The backbone of such methods rep-
+resent rational Krylov-type approaches together with the Sylvester matrix equation interpretation
+[1, 11, 20]. Apart from being computationally efficient and easy to implement, MM approaches have
+another advantage: they do not (necessarily) require access to a full-state realization of the original
+dynamical system. Hence, they can be viewed as data-driven methods. Here data are given by the
+moments of the system, i.e., samples of the underlying transfer function of the system (and of its
+derivatives) evaluated in a particular frequency range; for more details, we refer the readers to [5,8,41]
+and to Chapter 3 in [13]. The notion of a moment with respect to systems and control theory is related
+to the unique solution of a Sylvester matrix equation [20].
+The purpose of this note is twofold; first, we intend to review and to connect three important system
+theoretical model reduction methods based on interpolation that were introduced in the last 15 years:
+• The Loewner framework (LF) by Mayo and Antoulas from 2007 in [41];
+• The Astolfi framework (AF) from 2010 in [8];
+• The Adaptive Antoulas Anderson (AAA) algorithm by Nakatsukasa, Set´e and Trefethen from
+2018 in [42].
+Together, these three approaches were cited multiple times in various research publications, being
+arguably quite popular methods. However, until now, not too many connections between them were
+provided, neither in the automatic control, nor in the model reduction, or numerical analysis commu-
+nities. Together with the vector fitting algorithm (VF) in [31] (which is not based on interpolation,
+and is hence a purely optimization approach based on least-squares fitting), these methods repre-
+sent arguably the most prolific rational approximation schemes developed in the system and control
+community. However, VF is not the object of this study since it is not based on interpolation.
+The other scope of this note is to propose a new method that is based on the three methods
+enumerated above, and that addresses some of the shortcomings and challenges associated with them.
+Basically, the idea is to combine the attributes of each method, by following the steps below.
+• We make use of the order-revealing property of the LF (encoded by the rank of augmented
+Loewner matrices); additionally, the selection of interpolation points is done via a Loewner-CUR
+technique proposed in [38].
+• We utilize the elegant state-space parameterization of the LTI system proposed by the AF (after
+imposing k interpolation conditions); this is the backbone of the methods (we also show the
+connection between state-space forms and barycentric forms).
+• We use either the fitting step from AAA (that chooses free parameters to fit the un-interpolated
+data in a least square sense) or we impose pole placing (dominant poles are selected from those
+of the Loewner model); in both cases, a linear system of equations needs to be solved.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+3
+In what follows, we consider a multiple-input multiple-output (MIMO) linear time-invariant (LTI)
+system ΣL of dimension n described by the following system of differential equations:
+ΣL :
+�
+˙x(t) = Ax(t) + Bu(t),
+y(t) = Cx(t),
+(1)
+with x(t) ∈ Rn as the state variable, u(t) ∈ Rm as the control inputs, and y(t) ∈ Rp as the observed
+outputs. Here, we have that A ∈ Cn×n, B ∈ Cn×m and C ∈ Cp×n. The transfer (matrix) function of
+the LTI system is given by H(s) ∈ Cp×n, with s ∈ C, as
+H(s) = C(sIn − A)−1B.
+(2)
+It is to be noted that, for m = p = 1, the system becomes single-input single-output (SISO). We will
+sometime switch between MIMO and SISO formats while presenting the methods covered in this note,
+since the latter allows a more easy exposition for some of the results shown here.
+Let si ∈ C \ σ(A), where σ(A) denotes the spectrum of matrix A ∈ Cn×n, i.e., the set of its
+eigenvalues.
+The j-moment of system ΣL at si is given by ηj(si) =
+(−1)j
+j!
+�
+dj
+dsj H(s)
+�
+s=si, for any
+integer j ⩾ 1. The 0-moment is obtained by sampling the transfer function H(s) in (2) at si, i.e.,
+η0 = H(si). In this contribution, we restrict the analysis to matching 0-moments, i.e., samples of
+the transfer function H(s), and not of its derivatives. However, all methodologies shown here can be
+expected to cope with this as well. Moreover, in practice, inferring 0-moments from time-domain data
+is usually a more straight-forward task; this is performed by exciting the system with harmonic inputs,
+and by applying spectral transformations to the outputs. Additionally, the inference of derivative
+values (of the transfer function) is typically susceptible to perturbations and it is more challenging to
+attain, from a numerical point of view.
+The paper is structured in the following way; after the introduction session sets up the stage, we
+propose a survey of three established interpolation-based methods in Section 2. Then, the proposed
+methodologies are developed in Section 3, with emphasis on the one-step approach that combines
+optimal selection of interpolation points (chosen using CUR-DEIM) with LS fit on the rest of the
+data, and also the pole placement method in barycentric form that enforces dominant poles from the
+Loewner data-driven model. Then, Section 4 illustrates the numerical aspects of applying the methods
+discussed/proposed in the previous two sections to a variety of test cases (various models and data
+sets). Finally, Section 5 presents the conclusions and the outlook into future research.
+2. A survey of established methods
+In this section we discuss three established data-driven methods for rational approximation (AF, LF,
+and AAA, as mentioned in the previous section).
+The data are samples of the transfer function
+corresponding to the underlying dynamical system, measured on a particular frequency grid. In what
+follows, we mention some state-of-the-art methodologies used to measure such data, i.e., frequency
+response data. Typically, such measurements can be produced in practice from experiments conducted
+in scientific laboratories using carefully calibrated machines, called spectrum analyzers (SAs). In this
+category we mention swept-tuned spectrum analyzers, scalar network analyzers (SNAs), and vector
+network analyzers (VNAs).
+The SNA is an instrument that measures microwave signals by converting them to a DC voltage
+using a diode detector. In a VNA, information regarding both the magnitude and the phase of a
+microwave signal is extracted.
+While there are different ways to perform such measurements, the
+method employed by commercial products (such as the Anritsu series described in [18]) of VNAs is
+to down-convert the signal to a lower intermediate frequency in a process called harmonic sampling.
+This signal can then be measured directly by a tuned receiver. Compared to the SNA, the VNA is
+a more powerful analyzer tool. The major difference is that the VNA can also measure the phase,
+and not only the amplitude. With this property enforced, then so-called scattering parameters (or
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+4
+S-parameters) can be processed. These can be used for identifying forward and reverse transmission
+and reflection characteristics. More details can be found in [18].
+The harmonic balance method (or HBM) [43], is an established methodology in the field of elec-
+tromagnetics. The HBM is used in many (if not most) commercial radio-frequency (RF) simulation
+tools. This is due to the fact that it has certain advantages over other common methods, namely
+modified nodal analysis (MNA), which makes it more appropriate to use for stiff problems and circuits
+containing transmission lines, nonlinearities and dispersive effects. More details can be found in the
+survey paper [49].
+2.1. The one-sided moment-matching approach in [8]
+The framework introduced by Astolfi in [8] (referred to as AF throughout the paper) deals with the
+problem of model reduction by moment matching. Although classically interpreted as a problem of
+interpolation of points in the complex plane, it has instead been recast as a problem of interpolation
+of steady-state responses. In the following we briefly review its application to linear systems. It is to
+be noted that the AF was steadily extended and applied to different scenarios (including nonlinear
+dynamical systems, pole-zero placement, and least-squares fit) [33–35,45,53,55].
+The moments of a linear system can be characterized in terms of the solution of Sylvester equations.
+By using this observation, it has been shown that the moments are in one-to-one relation with the
+steady-state output response of the interconnection between a signal generator and the original linear
+system.
+In what follows, for simplicity of exposition, it is assumed that ΣL is a minimal system (both fully
+controllable and fully observable). For exact definitions on minimality, controllability, or observability
+of LTI systems, we refer the reader to [1].
+Let k ⩽ n and S ∈ Ck×k a non-derogatory matrix (for which the characteristic and minimal
+polynomials coincide) with σ(S) ∩ σ(A) = ∅ and R ∈ C1×k so that (S, R) is observable. Consider the
+signal generator system Σsg described by the equations
+Σsg :
+�
+˙ω(t) = Sω(t),
+u(t) = Rω(t).
+(3)
+Then, the explicit solution of (3) can be written as ω(t) = eStω(0). Hence, the control input is written
+as u(t) = ReStω(0). In addition, the eigenvalues of S are called interpolation points.
+For a linear system ΣL, and interpolation points si ∈ C \ σ(A), for i = 1, . . . , k, consider a non-
+derogatory matrix S ∈ Rk×k. It follows that there exists a one-to-one relation between the moments
+of the system ΣL and
+1. the matrix CΠ, where Π is the (unique) solution of the Sylvester equation AΠ + BR = ΠS,
+for any row vector R ∈ R1×k so that (R, S) is observable;
+2. the steady-state response of the output y of the interconnection of system ΣL and the system
+Σsg, for any R and ω(0) such that the triplet (R, S, ω(0)) is minimal.
+More precisely, let ∆ ∈ Rk be a column vector containing k free parameters (denoted here by
+δ1, δ2, . . . , δk, with δi ̸= 0, 1 ≤ i ≤ k). Then, as stated in [8], the family of linear time-invariant
+systems that interpolates the moments of system ΣL at the eigenvalues of matrix S, is given by
+ˆΣ∆ :
+� ˙ˆx(t) = (S − ∆R)
+�
+��
+�
+= ˆA
+ˆx(t) + ∆
+����
+= ˆB
+u(t),
+ˆy(t) = CΠ
+����
+= ˆC
+ˆx(t),
+(4)
+where the matrices S and R are as before and Π is the unique solution of the Sylvester equation
+AΠ + BR = ΠS. Additionally, the condition σ(S) ∩ σ(S − ∆R) = ∅ needs to be enforced. It is
+to be noted that the free parameters explicitly enter the vector ˆB = ∆, but also the matrix ˆA, as
+ˆA = S − ∆R. Finally, ˆC = CΠ has fixed entries.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+5
+The Sylvester matrix equation for the reduced-order system is written as ˆA ˆΠ+ ˆBR = ˆΠS. This can
+be explained by the fact that the reduced-order system matches the prescribed moments of the original
+system, hence the same format of the two equations. Without loss of generality, one can consider
+that the matrix ˆΠ is the identity matrix, i.e.
+ˆΠ = Ik (this can be achieved by applying similarity
+transformations). By replacing this value into the reduced-dimension Sylvester matrix equation above,
+the formula ˆA = S − ∆R directly follows.
+Afterwards, the free parameters collected in the vector ∆ can be chosen in order to enforce or impose
+additional conditions as mentioned in [8], such as: matching with imposing additional k interpolation
+conditions, matching with prescribed eigenvalues, matching with prescribed relative degree, matching
+with prescribed zeros, matching with a passivity constraint, matching with L2-gain, or matching with
+a compartmental constraint.
+An important aspect of the AF is the characterization of all, i.e., infinitely many families of reduced-
+order models that satisfy k prescribed interpolation conditions. This is done by explicitly computing
+such parameterized models, for which the free parameters are the variables entering in the vector ∆.
+The main parameterization developed here will be used as a “backbone” of the methods developed in
+Section 3.
+As stated in the original paper, the main advantage of the AF (characterization of moments in
+terms of steady-state responses) is that it allows the definition of moments for systems which do not
+admit a clear/immediate representation in terms of transfer function(s). Hence, the author provides
+as examples the case of linear time-varying systems, and the case of nonlinear systems. Moreover,
+it is stated in [8] that one disadvantage of the framework is that it requires the existence of steady-
+state responses. Consequently, the original system has to be (exponentially) stable. However, in most
+practical applications, this is a realistic requirement.
+2.2. The Loewner framework in [41]
+In this section we present a short summary of the Loewner framework (LF), as introduced in [41]. It
+is to mentioned that LF has its roots in the earlier work of [4], and that LF can be considered to be a
+double-sided moment-matching approach (as opposed to AF, which is one-sided).
+For a tutorial paper on LF for LTI systems, we refer the reader to [7], and for a recent extension
+that uses time-domain data, we refer the reader to [48]. The Loewner framework has been recently
+extended to certain classes of nonlinear systems, such as bilinear systems in [6], and quadratic-bilinear
+(QB) systems in [24], but also to linear parameter-varying systems in [28]. Additionally, issues such
+as stability preservation or enforcement, or passivity preservation, were tackled in the LF in [23, 29],
+for the former, and in [2,12], for the latter.
+The LF is based on processing frequency-domain measurements D = {(ωℓ, H(ωℓ)) , ℓ = 1, . . . , N}
+(with ωℓ ∈ R for 1 ≤ ℓ ≤ N) corresponding to evaluations of the transfer function of the underlying
+(unknown/hidden) dynamical system.
+The interpolation problem is formulated as shown below (for convenience of exposition, we show
+here only the SISO formulation). We are given data nodes and data points in the set D, partitioned
+into two disjoint subsets DL and DR, with DL ∪ DR = D and k + q = N, as
+right data : DL = {(λj, H(λj)) , j = 1, . . . , k}, and,
+left data : DR = {(µi, H(µi)) , i = 1, . . . , q},
+(5)
+and we seek to find a rational function ˆH(s), such that the following interpolation conditions hold:
+ˆH(µi) = H(µi) := vi,
+ˆH(λj) = H(λj) := wj.
+(6)
+The Loewner matrix L ∈ Cq×k and the shifted Loewner matrix Ls ∈ Cq×k play an important role in
+the LF, and are given by
+L(i,j) = vi − wj
+µi − λj
+,
+Ls(i,j) = µivi − λjwj
+µi − λj
+,
+(7)
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+6
+while the data vectors V ∈ Cq, WT ∈ Ck are given by
+V(i) = vi,
+W(j) = wj, for i = 1, . . . , q, j = 1, . . . , k.
+(8)
+Moreover, the following Sylvester matrix equations ([1, Ch. 6]) are satisfied by the Loewner and shifted
+Loewner matrices (here, 1q =
+�
+1
+· · ·
+1
+�T ∈ Cq)
+�
+ML − LΛ = V1T
+k − 1qW,
+MLs − LsΛ = MV1T
+k − 1qWΛ,
+(9)
+where M = diag(µ1, . . . , µq) and Λ = diag(λ1, . . . , λk) are diagonal matrices. The following relation
+holds true
+Ls = LΛ + V1T
+k = ML + 1qW.
+(10)
+The unprocessed Loewner surrogate model, provided that k = q, is composed of the matrices
+ˆE = −L,
+ˆA = −Ls,
+ˆB = V,
+ˆC = W,
+(11)
+and if the pencil (L, Ls) is regular, then the function ˆH(s) satisfying the interpolation conditions in
+(6) can be explicitly computed in terms of the matrices in (11), as ˆH(s) = ˆC(sˆE − ˆA)−1 ˆB.
+In practical applications (when processing a fairly large number of measurements), the pencil (Ls, L)
+is often singular. Hence, a post-processing step is required for the Loewner model in (11). In such
+cases, one needs to perform a singular value decomposition (SVD) of augmented Loewner matrices, to
+extract the dominant features and remove inherent redundancies in the data. By doing so, projection
+matrices X, Y ∈ Ck×r are obtained, as left, and respectively, right truncated singular vector matrices:
+[L Ls] = YS(1)
+r
+˜X
+H � L
+Ls
+�
+= ˜YS(2)
+r XH,
+(12)
+where S(1)
+r , S(2)
+r
+∈ Rr×r,
+Y ∈ Ck×r, X ∈ Cq×r, ˜Y ∈ C2q×r, ˜X ∈ Cr×2k. The truncation index r
+can be chosen as the numerical rank (based on a tolerance value τ > 0) or as the exact rank of the
+Loewner pencil (in exact arithmetic), depending on the application and data size. More details can be
+found in [7].
+The system matrices corresponding to a projected Loewner model of dimension r can be computed
+as follows:
+˜E = −XHLY,
+˜A = −XHLsY,
+˜B = XHV,
+˜C = WY.
+We note that MIMO extensions of the LF were already proposed in the original contribution [41].
+There, a tangential interpolation framework is considered. Instead of imposing interpolation of full
+p × m blocks, the authors prefer to interpolate the original transfer matrix function samples along
+certain vectors (or tangential directions). We also note that a first attempt of re-interpreting the LF
+in [41] as a one-sided method was made in [25]. In the latter, the main difference to the classical work
+in [4] was that a compression of the left (un-interpolated) data set was enforced. However, in [25], it
+was still unclear how to split the data, i.e., what the right data set should be (where interpolation is
+enforced). Finally, it is to be noted that the choice of interpolation points is crucial in the LF. An
+exhaustive study of different choices was proposed in [37], while a greedy strategy was proposed in
+[17], for scenarios in which limited experimental data are available.
+2.3. The AAA algorithm in [42]
+The AAA algorithm introduced in [42] is an adaptive and iterative extension of the interpolation
+method based on Loewner matrices, originally proposed in [4]. The main steps are as follows
+1. Express the fitted rational approximants in a barycentric representation, which represents a
+numerically stable way of expressing rational functions [15].
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+7
+Algorithm 1 The AAA algorithm.
+Require: A (discrete) set of sample points Γ ⊂ C with N points, function f (or the evaluations of f
+on the set Γ, i.e., the sample values), and an error tolerance ϵ > 0.
+Ensure: A rational approximant rn(s) of order (n, n) displayed in a barycentric form.
+1: Initialize j = 0, Γ(0) ← Γ, and r−1 ← N −1 �N
+i=1 f(γi).
+2: while |f(s) − rj−1(s)| > ϵ do
+3:
+Select a point zj ∈ Γ(j) for which |f(s) − rj−1(s)| attains a maximal value, where for j ≥ 1, it
+follows:
+rj−1(s) :=
+�j−1
+�
+k=0
+ω(j−1)
+k
+s − zk
+�−1 �j−1
+�
+k=0
+ω(j−1)
+k
+fk
+s − zk
+�
+.
+(13)
+4:
+if |f(zj) − rj−1(zj)| ≤ ε then
+5:
+Return rj−1.
+6:
+else
+7:
+fj ← f(zj) and Γ(j+1) ← Γ(j) \ {zj}.
+8:
+end if
+9:
+Find the weights ω(j) = [ω(j)
+0 , . . . , ω(j)
+j ] by solving a least squares problem over z ∈ Γ(j+1)
+j
+�
+k=0
+ω(j)
+k
+s − zk
+f(s) ≈
+j
+�
+k=0
+ω(j)
+k fk
+s − zk
+⇔
+�
+j
+�
+k=0
+f(s) − fk
+s − zk
+�
+ω(j)
+k
+≈ 0 ⇔ L(j)ω(j) = 0.
+(14)
+The solution of (14) is given by the (j + 1)th right singular vector of the Loewner matrix
+L(j) ∈ C(N−j−1)×(j+1).
+10:
+j ← j + 1.
+11: end while
+2. Select the next interpolation (support) points via a greedy scheme; basically, interpolation is
+enforced at the point where the (absolute or relative) error at the previous step is maximal.
+3. Compute the other variables (the so-called barycentric weights) in order to enforce least squares
+approximation on the non-interpolated data.
+In recent years, the AAA algorithm has proven to be an accurate, fast, and reliable rational ap-
+proximation tool with a fairly large range of applications. Here, we will mention only a few: nonlinear
+eigenvalue problems [39], MOR of parameterized linear dynamical systems [16], MOR of linear sys-
+tems with quadratic outputs [26], rational approximation of periodic functions [10], representation of
+conformal maps [22], rational approximation of matrix-valued functions [27], or signal processing with
+trigonometric rational functions [60]. The procedure is sketched in Algorithm 1.
+It is to be mentioned that a modified version of AAA that enforces real-valued and strictly-
+proper rational appoximants was recently proposed in [30]. There, the format of the function in (13)
+was modified by inserting a 1 into the denominator, as follows
+˜rj(s) :=
+�
+1 +
+j−1
+�
+k=0
+ω(j−1)
+k
+s − zk
+�−1 �j−1
+�
+k=0
+ω(j−1)
+k
+fk
+s − zk
+�
+.
+(15)
+Consequently, the equation in (14) becomes L(j)ω(j−1) = −f(j−1), where the vector f(j−1) ∈ Cj is
+given by f(j−1) =
+�f0
+f2
+· · ·
+fj−1
+�T. It is to be noted that ˜rj(s) in (15) is theoretically a rational
+approximant of order (j − 1, j), if we do not take into account pole/zero cancellations or any other
+zero cancellations of coefficients in the numerator or denominator.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+8
+3. The proposed methodologies
+3.1. Skeleton of the main methods
+Similar to the methods reviewed in Section 2 we want to find an LTI system with a transfer function of
+the structure (1) that interpolates data provided as measurements H (si) , i = 1, . . . , k of the transfer
+function of the original system. We can directly put together an LTI parametrized model of dimension
+r = km, having km2 degrees of freedom with transfer function
+ˆH(s) = ˆC(sIr − ˆA)−1 ˆB,
+(16)
+with the underlying data concatenated to
+ˆC =
+�H(λ1)
+· · ·
+H(λk)�
+∈ Cp×r,
+(17)
+a matrix of weights ˆ
+Wi
+ˆB =
+�
+ˆ
+W
+H
+1
+· · ·
+ˆ
+W
+H
+k
+�H
+∈ Cr×m,
+(18)
+and ˆA ∈ Cr×r formed from a diagonal matrix populated with the interpolation points λi disturbed by
+ˆB, such that
+ˆA = Λ − ˆBR = diag (λ1, . . . , λk) ⊗ Im − ˆB
+�
+1T
+r ⊗ Im
+�
+.
+(19)
+Making use of the Woodbury matrix identity and denoting Λs = sIkm − Λ, the transfer function (16)
+can be rewritten as
+ˆH(s) = ˆCΛ−1
+s
+ˆB
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+.
+(20)
+A complete derivation of (20) is given in Appendix A.1.
+In the single-input single-output case (m = p = 1, hence r = k), the barycentric weights reduce to
+scalars and the matrices for a ROM of structure (16) are given by
+ˆA = Λ − ˆBR ∈ Ck×k,
+ˆB =
+� ˆw1
+· · ·
+ˆwk
+�T ∈ Ck×1,
+ˆC =
+�H (λ1)
+· · ·
+H (λk)�
+∈ C1×k.
+(21)
+By inserting the formulae in (21) into (20), and using the notation hi := H (λi), leads to
+ˆCΛ−1
+s
+ˆB =
+k
+�
+i=1
+ˆwihi
+s − λi
+,
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+=
+1
+1 + �k
+i=1
+ˆwi
+s − λi
+.
+(22)
+Hence, the transfer function of the model in (21) is given in barycentric representation by
+ˆH(s) =
+�k
+i=1
+ˆwihi
+s − λi
+1 + �k
+i=1
+ˆwi
+s − λi
+.
+(23)
+This can be performed analogously for a multi-input multi-output case (m = p, r = km). The first
+part of (20) becomes
+ˆCΛ−1
+s
+ˆB =
+�H(λ1)Im(s − λ1)−1
+· · ·
+H(λk)Im(s − λk)−1�
+�
+��
+ˆ
+W1
+...
+ˆ
+Wk
+�
+�� =
+k
+�
+i=1
+H(λi) ˆ
+Wi
+s − λi
+,
+(24)
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+9
+the second part
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+=
+�
+�
+�
+�Im +
+�Im
+· · ·
+Im
+�
+�
+��
+Im(s − λ1)−1
+...
+Im(s − λk)−1
+�
+��
+−1 �
+��
+ˆ
+W1
+...
+ˆ
+Wk
+�
+��
+�
+�
+�
+�
+−1
+=
+�
+Im +
+k
+�
+i=1
+ˆ
+Wi
+s − λi
+�−1
+.
+(25)
+Consequently, the transfer function in (20) has also a barycentric form given by
+ˆH(s) =
+� k
+�
+i=1
+H(λi) ˆ
+Wi
+s − λi
+� �
+Im +
+k
+�
+i=1
+ˆ
+Wi
+s − λi
+�−1
+.
+(26)
+The transfer function is defined by the choice of the interpolation points and of the weights. The
+interpolation points can be chosen as dominant parts of the available data or based on their location in
+the frequency spectrum. The weights can be computed such that the data which are not interpolated,
+are approximated in an optimal way. Alternatively, the weights can be chosen to enforce poles at
+specific locations. In the following, we show different strategies for both choices.
+3.2. Automatic choice of interpolation points
+The approximation quality of a surrogate model of the form (16) is greatly influenced by the choice of
+the interpolation points λ. This choice is not always obvious, so automatic strategies are frequently
+employed. The Loewner framework uses the SVD to identify dominant subsets of the available data
+to enforce interpolation on. Alternatively, the AAA algorithm uses a greedy scheme to minimize the
+error between surrogate and original data. Another approach, originally introduced by [37], makes use
+of the CUR decomposition to extract interpolation points from a relevant subset of the available data.
+The CUR decomposition approximates a matrix A by a product of three low-rank matrices ˇA =
+ˇC ˇU ˇR, where ˇC and ˇR represent subsets of the columns respectively rows of A [40,56]. In our case the
+three matrices are only a byproduct, we are more interested in the interpolation points λ and µ that
+are associated to the columns and rows extracted as ˇC and ˇR. In combination with the skeleton for a
+realization described in Section 3.1, Algorithm 2 computes a surrogate model approximating a set of
+given transfer function data. We use the algorithm from [56] to compute the CUR decomposition and
+thus identify dominant parts of the original data set and their corresponding left and right interpolation
+points. Contrary to [37], we decompose the original Loewner matrix L rather than the augmented
+Loewner matrices
+�L
+Ls
+�
+and
+�
+LH
+LH
+s
+�H. Using all interpolation points obtained from the CUR
+decomposition would introduce redundant data into the surrogate. Therefore we choose only a subset
+of the interpolation points: either only the left points, only the right points, or every other entry from
+a concatenated and sorted vector of left and right points. Together with the data associated to the
+chosen interpolation points they are used to populate a rectangular Loewner matrix. We now need
+to compute weights for barycentric interpolation as described in the following section. After having
+obtained the weights, a surrogate model (16) can be computed from (17)–(19).
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+10
+Algorithm 2 LS-Loewner with CUR.
+Require: Transfer function samples {H (si)}N
+i=1, corresponding sampling points Ξ = {si}N
+i=1.
+Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.
+1: Partition data and compute Loewner matrix L as in (7).
+2: Compute CUR decomposition, such that L = ˇC ˇU ˇR with ˇC ∈ CN×k, ˇR ∈ Ck×N.
+3: Obtain interpolation points {λi}k
+i=1 , {µi}k
+i=1 corresponding to the columns and rows in ˇC, ˇR.
+4: Postprocess interpolation points to obtain ν = {ν}k
+i=1 and χ = Ξ \ ν.
+5: Populate a rectangular Loewner matrix L(i,j) = H(χi)−H(νj)
+χi−νj
+.
+6: Compute the weights Ω = −L† �
+H (ν1)H
+· · ·
+H (νk)H�H
+, where L† is the pseudo-inverse of L and
+Ω =
+�
+ˆ
+W
+H
+1
+· · ·
+ˆ
+W
+H
+k
+�H
+.
+7: Compute ˆA, ˆB, ˆC with (17)–(19).
+3.3. Computing the barycentric weights
+3.3.1. Least-squares approach
+The matrix-valued weights ˆ
+Wi can be computed similarly to AAA [27] by solving the minimization
+problem
+min
+ˆ
+Wi
+h
+�
+j=1
+�
+�
+� k
+�
+i=1
+H(λi) ˆ
+Wi
+sj − λi
+� �
+Im +
+k
+�
+i=1
+ˆ
+Wi
+sj − λi
+�−1
+− H(sj)
+�
+�
+2
+.
+(27)
+This solution can, for example, be obtained from an optimization in least-squares sense. The weights
+for the SISO case are computed analogously. Here, the matrix-values weights and transfer function
+values reduce to scalars.
+3.3.2. Pole placement
+The next step would be to take advantage of the degrees of freedom in the vector ˆB from (21), so
+that the ROM thus constructed has particular (stable) poles [21, 35, 46]. These will be denoted with
+ζ1, ζ2, . . . , ζk. The following derivations assume a SISO model. To enforce that this happens, we need
+to make sure that the matrix ζjIk − ˆA loses rank for all 1 ≤ j ≤ k. In what follows, we show how to
+enforce this property in an elegant, straightforward way. Remember that the transfer function of the
+parameterized AF model is given by:
+ˆH(s) =
+�k
+i=1
+ˆwihi
+s − λi
+1 + �k
+i=1
+ˆwi
+s − λi
+= N(s)
+D(s).
+(28)
+Now, let’s say we would like this transfer function to have k poles at the selected values ζj’s. Clearly,
+the condition is D(ζj) = 0 and hence we need to enforce:
+1 +
+k
+�
+i=1
+ˆwi
+ζj − λi
+= 0, ∀1 ≤ j ≤ k ⇔ Cζ,λ ˆB = −1k ⇔ ˆB = −C−1
+ζ,λ1k,
+(29)
+where Cζ,λ is a Cauchy matrix defined by: (Cζ,λ)i,j =
+1
+ζi−λj . Details on how to obtain the above
+expression by following the procedure in [3] are given in Appendix A.2. We note that placing poles
+is a difficult numerical problem which requires the inversion of a Cauchy matrix, which is highly
+ill-conditioned, by nature.
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+11
+Algorithm 3 Loewner framework with pole placement (LFPP).
+Require: Transfer function samples {H (si)}N
+i=1, corresponding sampling points Ξ = {si}N
+i=1, loca-
+tions for poles ζ = {ζi}k
+i=1, interpolation points λ = {λi}k
+i=1.
+Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.
+1: Compute ΣD from {H (si)}N
+i=1 and {si}N
+i=1 using the Loewner framework (cf. Section 2.2).
+2: ˆC ←
+�HD (λ1)
+· · ·
+HD (λk)�
+.
+3: ˆB ← −C−1
+ζ,λ1r.
+4: ˆA ← diag (λ1, . . . , λk) − ˆB1T
+k.
+Instead of doing this, we could solve Cζ,λ ˆB = −1r, without inverting the Cauchy matrix explicitly,
+i.e., by solving a linear systems of equations. Algorithm 3 summarizes this procedure in a data-driven
+context. The required underlying model is obtained from a set of transfer function evaluations by
+applying the Loewner framework. The method is illustrated for SISO systems, but can readily be
+extended to the MIMO case.
+3.4. Automatic choice of poles and interpolation points
+A reasonable choice of poles and interpolation points for Algorithm 3 is not always readily available,
+but the approximation of the surrogate is heavily influenced by this choice. In the following, we show
+an extension to Algorithm 3 which computes a surrogate model (16) without requiring sets of poles
+and interpolation points as input parameters. Algorithm 4 sketches the skeleton of such automatic
+algorithm. Similar to Algorithm 3 it employs the Loewner framework to obtain a realization of a
+surrogate interpolating the provided data.
+Subsequently, a generalized eigendecomposition of the
+Loewner realization of the original data is computed to find suitable locations for poles. From this
+is is possible to compute the dominance of all eigenvalues; for details, see, e.g. [51]. The algorithm
+now chooses the k most dominant eigenvalues as poles to enforce in the surrogate. It should be noted
+that only eigenvalues with negative real parts should be considered, if the stability of the surrogate is
+important. The required interpolation points can now be chosen similar to Algorithm 2 by computing
+a CUR decomposition and using the interpolation points associated to the rows or columns of the
+decomposition as interpolation points for the new surrogate.
+The approximation of dominant poles of the underlying model from data is less robust if the transfer
+function samples are disturbed by noise. This leads to a reduced approximation quality. For a better
+performance if applied to noisy data, Algorithm 4 can be modified as follows: To obtain poles which
+should be enforced, first choose manually the most prominent features in the transfer function, e.g.
+peaks, which should be approximated by the surrogate model.
+Now choose the eigenvalues which
+imaginary parts are closest to the frequencies, where the chosen features of the transfer function are
+located. The CUR decomposition also fails at extracting the most dominant rows and columns of
+the Loewner matrix if noisy data is assessed. Therefore another heuristic is employed to choose the
+interpolation points: Use the value si which corresponds to the lowest amplitude of the transfer function
+between the locations of two enforced poles. This leads to reasonable approximations, especially for
+lightly damped systems. Other approaches include choosing simply the middle between the location
+of two poles or specifying an offset between pole and interpolation point location.
+4. Numerical results
+In the following, we demonstrate the methods discussed in Section 3 by applying them on three
+benchmark examples available from the MOR-Wiki1:
+1http://modelreduction.org
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+12
+Algorithm 4 Loewner framework with automatic pole placement (LFaPP).
+Require: Transfer function samples {H (si)}N
+i=1, corresponding sampling points Ξ = {si}N
+i=1.
+Ensure: Surrogate model ˆH(s) = ˆC(sIr − ˆA)−1 ˆB.
+1: Compute ΣD from {H (si)}N
+i=1 and {si}N
+i=1 using the Loewner framework (cf. Section 2.2).
+2: Compute the generalized eigenvalue decompositions AX = EXα and YHA = αYHE for the
+matrices of right and left eigenvectors X, Y and the matrix of eigenvalues α = diag (α1, . . . , αn).
+3: Compute eigenvalue dominance di = |CY(:,i)αiX(:,i)HB|
+|ℜ(αi)|
+, i = 1, . . . , n and sort α accordingly
+4: Set ζ to the k most dominant eigenvalues.
+5: Compute CUR decomposition of L.
+6: Set λ to the k right or left interpolation points corresponding to the CUR decomposition.
+7: Compute surrogate as in Algorithm 3.
+ISS This system models the structural response of the Russian Service Module of the International
+Space Station (ISS) [52]. The model has n = 270 states, m = 3 inputs, and p = 3 outputs. The
+dataset used for the computations contains transfer function measurements at 400 logarithmically
+distributed points in the range
+�
+10−1, 102�
+· ı. The model is also part of the SLICOT benchmark
+collection [44].
+Flexible aircraft This system models lift and drag along the flexible wing of an aircraft. The system
+matrices are not available, we only have access to a dataset of 420 transfer functions samples
+at linearly distributed frequencies between 0.1 and 42.0 Hz. The original dataset has one input
+(the gust disturbance) and 92 outputs. For the following experiments, we choose the 91st output
+which corresponds to the first flexible mode [50]. The dataset is available from [57].
+Sound transmission This system models the sound transmission through a system of two brass plates
+with an air enclosure between them. The transfer function measures the sound pressure in an
+adjacent acoustic cavity. The geometry is based on [32]; the data—transfer function evaluations
+at 1000 linearly-distributed frequency values between 1 and 1000 Hz—is available from [9].
+We note that no tangential interpolation (as described in [41]) is applied for the MIMO model.
+Instead, the Loewner matrices are constructed in a block-wise manner. The case of tangential inter-
+polation, within the proposed approaches in this note, will be investigated in future works.
+We enforce realness of all surrogate models (all matrices contain only real entries) by applying the
+transformation described in [7]. For this, all data must be available in complex conjugate pairs. The
+required transformation matrix is given by
+J = Iℓ ⊗
+� 1
+√
+2
+� Im
+Im
+−ıIm
+ıIm
+��
+,
+(30)
+with ℓ = k
+2 and the real-valued quantities are obtained from ˆA
+(ℜ) = J ˆAJH, ˆB
+(ℜ) = J ˆB, and ˆC
+(ℜ) =
+ˆCJH.
+For some of the experiments we add artificial noise to the measurements, in order to obtain perturbed
+data. The modified measurements are given by
+ˇH (si) = H (si) (1 + Zi) , i = 1, . . . , n,
+(31)
+where Zi ∈ C is the ith sample drawn from a set of random numbers Z ∼ CN
+�
+µ, σ2�
+following a
+complex normal distribution with mean µ and standard deviation σ2. Here, the real and imaginary
+parts of Z are independent normally distributed variables [19].
+We assess the approximation error of the surrogate models with an approximated L∞ norm, because
+many surrogates have unstable poles and hence, the H∞ can not be computed. For a given reduced
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+13
+order r, the L∞ error in the considered frequency range ω ∈ [ωmin, ωmax] is approximated by
+ε(r) =
+max
+ω∈[ωmin,ωmax]
+���H(ωı) − ˆHr(ωı)
+���
+2
+max
+ω∈[ωmin,ωmax] ∥H(ωı)∥2
+≈
+���H − ˆHr
+���
+L∞
+∥H∥L∞
+.
+(32)
+Note that strategies to post-process surrogates to obtain stable models have been studied in [29].
+The numerical experiments have been conducted on a laptop equipped with an AMD Ryzen™
+7 PRO 5850U and 12 GB RAM running Linux Mint 21 as operating system. All algorithms have been
+implemented and run with MATLAB R2021b Update 2 (9.11.0.1837725).
+Code and data availability
+The data that support the findings of this study are openly available in Zenodo at
+doi:10.5281/zenodo.7490158
+under the BSD-2-Clause license, authored by Quirin Aumann and Ion Victor Gosea.
+4.1. Case of exact measurement data
+In the following, we compare the performance of the new approach LS-Loewner to the following estab-
+lished strategies:
+• Loewner-SVD: Truncate Loewner matrices populated with the complete dataset to order r using
+an SVD [7].
+• Loewner-CUR: Construct a purely interpolatory model of order r using all data points chosen by
+the CUR decomposition, similar to [37].
+• Modified AAA: Apply the strictly-proper variant of AAA [29] to the complete dataset to compute
+a reduced-order model of size r.
+We first consider the original MIMO ISS example and a SISO variant where we select the first input
+and output, respectively, from the MIMO system. To evaluate the overall performance of the different
+methods related to the size of a surrogate model, we compute the approximated L∞ errors for models
+with orders 6 ≤ r ≤ 60. The approximation error versus the dimension of the respective surrogate
+model is depicted in Figure 1 for all four methods.
+Since tangential interpolation was not employed here, the order of the MIMO surrogates rises by m
+for each additional interpolation point, i.e., r = km. This explains the lower accuracy of the MIMO
+surrogate. For the maximum reduced order r = 60, k = 20 interpolation points are considered. The
+errors of the SISO surrogates for r = 20, i.e., k = 20, is in a similar range as in the MIMO case. The
+SISO surrogates reach similar levels of approximation for all employed methods. In the MIMO case,
+Loewner-SVD performs best. This can be explained by the following observation: the other methods
+always consider the complete transfer function measurement H(λi) ∈ Cp×m per interpolation point,
+while Loewner-SVD extracts only the r most dominant singular vectors for projection, regardless of
+to which interpolation point they belong to. In turn, the other methods also consider probably less
+important parts of the data as long as one input/output combination of the respective sample is
+relevant for approximation. It can also be noted that LS-Loewner and Loewner-CUR perform very
+similar. This was expected, as both methods rely on the same interpolation points.
+All four methods are now employed to compute a surrogate model of size r = 108 to approximate
+the transfer function of the flexible aircraft model. The size of the surrogate model is determined by
+truncating all singular values τ < 1·10−6 of an underlying Loewner matrix.
+The transfer functions of all resulting models and their respective relative errors are given in Figure 2.
+Again, all methods succeed in computing a sufficiently accurate surrogate. However, the approximation
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+14
+10
+20
+30
+40
+50
+60
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Reduced order r
+L∞ error
+SISO
+10
+20
+30
+40
+50
+60
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Reduced order r
+L∞ error
+MIMO
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+Figure 1: The approximated L∞ errors of reduced-order models of order r computed from the ISS
+data. Left: SISO with first input and output, respectively. Right: MIMO with three inputs
+and three outputs m = p = 3 (the number of interpolation points is k = r
+m).
+quality of Loewner-CUR is noticeably worse than that of the other three methods. Given that both
+Loewner-CUR and LS-Loewner use the same interpolation points, the weights computed from the least
+squares problem show a better performance compared to the partitioning approach used in Loewner-
+CUR.
+4.2. Perturbed measurement data
+Analyzing measurement data perturbed by noise is a challenging task for interpolatory methods, such
+as the Loewner framework and the AAA algorithm (as pointed out in, e.g., [27]). In this experiment
+we investigate the effect of noise to the performance of the four methods described above and show,
+how enforcing poles and/or interpolation points can increase the approximation quality. In the first
+experiment we consider transfer function data from the ISS model perturbed by noise with mean µ = 0
+and standard deviation σ2 = 0.15. We employ LFaPP and enforce poles at ı[.77, 2, 4, 5.6, 9.33, 37.9]
+near peaks of the transfer function. The resulting real-valued surrogate model has order r = 12. The
+transfer functions of the surrogate model with enforced poles and reduced models computed from the
+same noisy data with LS-Loewner, Loewner-SVD, Loewner-CUR, and Modified AAA are given in Figure 3.
+Enforcing the poles near peaks in the transfer function of the underlying data allows the surrogate
+to capture the behavior of the original data in a wider frequency range than applying LS-Loewner,
+Loewner-SVD, and Loewner-CUR. The choice of the locations, in which vicinity the poles should be
+chosen is, however, not automatized. Figure 3 also shows the relative errors of all surrogate models
+referenced to the original data without noise. While the enforced poles all have a negative real part,
+the models computed from the variants of the LF and AAA exhibit unstable eigenvalues. Thus, pole
+placement can be seen also as a means to enforce stability of the surrogate models. Alternatively,
+a post-processing step can be added to enforce stable models (for both LF and AAA methods), as
+performed in [29].
+We now evaluate the performance of the algorithms by applying them to heavily distorted trans-
+fer function measurements of the sound transmission problem. Noise with a standard deviation of
+σ2 = 0.25 is considered and three algorithms are employed to compute surrogates: Loewner-SVD,
+LFPP (Algorithm 3), and LFaPP (Algorithm 4). We also test the modifications to LFaPP described in
+Section 3.4. These results are denoted by “LFaPP mod.”. For LFPP we enforce poles at the eigenval-
+ues of the underlying Loewner model which imaginary parts are near 2πı [72, 189, 392, 401, 706, 856].
+These locations correspond to characteristic peaks in the transfer function. Further, we choose the in-
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+15
+5
+10
+15
+20
+25
+30
+35
+40
+10−4
+10−2
+100
+Magnitude
+Original data
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+5
+10
+15
+20
+25
+30
+35
+40
+10−12
+10−9
+10−6
+10−3
+100
+Frequency [Hz]
+Relative error
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+Figure 2: Transfer function (top) and relative pointwise errors (bottom) for reduced-order models of
+size r = 108 for the aircraft model. The error is plotted only at frequencies which do not
+coincide to interpolation points of the respective method.
+terpolation points at 2πı [138, 339, 369, 569, 712, 954], which lie at the dips between the enforced poles.
+Loewner-SVD and LFaPP do not require input parameters in addition to the measured data. Figure 4
+shows the transfer function of the resulting surrogate models in comparison to the original and noisy
+underlying data. It can be observed, that the automatic approaches Loewner-SVD and LFaPP (mod.)
+cannot approximate the transfer function well after the first two peaks, i.e., for frequencies higher than
+200 Hz, while LFPP approximates the original data over the complete frequency range with decent
+accuracy. The importance of reasonable interpolation points can be seen in the difference of LFPP and
+LFaPP mod., which have the same poles. It should be noted that the surrogate model computed by
+Loewner-SVD has two unstable poles while the other three surrogate models are stable. It is, however,
+not always clear a priori how to choose the poles and interpolation points for LFPP in order to achieve
+the best approximation quality possible. In this example, the noise level is too high for one of the
+automatic approaches to yield reasonable dominant interpolation points or poles.
+5. Conclusion and outlook
+In this contribution, we have proposed an extensive study of interpolation-based data-driven ap-
+proaches for approximating the response of linear dynamical systems.
+All methods require input
+and output data, i.e., transfer function measurements, while direct access to the system operators or
+the states is not required. We showed different approaches how to achieve compact surrogate models
+approximating the input/output behavior of the original system and how to ensure various properties
+of the surrogate models, such as stability. Strategies how to work with noisy measurement data have
+also been addressed.
+A natural extension of the framework described here is to apply the ideas of tangential interpolation
+as a means of modeling a MIMO system from data. Here, the tangential directions need to be incorpo-
+rated in the parameterized one-sided realization. Further topics include enforcing different structures
+of the original model in the surrogate model, e.g., second-order or delay structures. It would also be
+interesting to study the possibility of placing certain stable poles while achieving interpolation in a
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+16
+10−1
+100
+101
+102
+10−5
+10−3
+10−1
+Magnitude
+Noisy data
+Original data
+LFaPP
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+10−1
+100
+101
+102
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+Frequency
+Relative error
+Noise
+LFaPP
+LS-Loewner
+Loewner-SVD
+Loewner-CUR
+Modified AAA
+Figure 3: Transfer function of a surrogate with enforced poles compared to the noisy and original
+transfer function values. The transfer function of a model r = 12 computed from Loewner-
+SVD is given for reference.
+least-squares sense. Application cases for the proposed methodology could include damping optimiza-
+tion. Here, a family of parameterized interpolants could be used to find optimal positions for viscous
+dampers in a structural system.
+A. Appendix
+A.1. The Woodbury matrix identity
+We can expand the right part of (19), such that:
+ˆA = Λ − ˆBR ⇒ sIkm − ˆA = sIkm − Λ
+�
+��
+�
+ˆ
+M
++
+�
+��
+ˆ
+W1
+...
+ˆ
+Wk
+�
+��
+� �� �
+ˆU
+�Im
+�
+����
+ˆT
+�Im
+· · ·
+Im
+�
+�
+��
+�
+ˆV
+.
+(33)
+The Woodbury matrix identity is as follows:
+�
+ˆM + ˆU ˆT ˆV
+�−1
+= ˆM
+−1 − ˆM
+−1 ˆU
+�
+ˆT
+−1 + ˆV ˆM
+−1 ˆU
+�−1 ˆV ˆM
+−1,
+where ˆM, ˆU, ˆT and ˆV are conformable matrices: ˆM is n × n, ˆT is k × k, ˆU is n × k, and ˆV is k × n.
+This can be derived using blockwise matrix inversion. By denoting with Λs = sIkm − Λ, then the first
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+17
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1,000
+10−6
+10−2
+102
+Magnitude
+Noisy data
+Original data
+Loewner-SVD
+LFPP
+LFaPP
+LFaPP mod.
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1,000
+10−3
+10−2
+10−1
+100
+101
+102
+103
+Frequency [Hz]
+Relative error
+Noise
+Loewner-SVD
+LFPP
+LFaPP
+LFaPP mod.
+Figure 4: Transfer function (top) and relative pointwise errors (bottom) as well as the added noise for
+reduced-order models of size r = 12 for the aircraft model.
+transfer function of the fitted model is written:
+ˆH(s) = ˆC
+�
+sI3m − ˆA
+�−1 ˆB = ˆC
+�
+Λs + ˆU ˆV
+�−1 ˆB
+= ˆCΛ−1
+s
+ˆB − ˆCΛ−1
+s
+ˆU
+�
+Im + ˆVΛ−1
+s
+ˆU
+�−1 ˆVΛ−1
+s
+ˆB
+= ˆCΛ−1
+s
+ˆB − ˆCΛ−1
+s
+ˆB
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+RΛ−1
+s
+ˆB
+= ˆCΛ−1
+s
+ˆB
+�
+Im −
+�
+Im + ˆX
+�−1 ˆX
+�
+= ˆCΛ−1
+s
+ˆB
+�
+Im + ˆX
+�−1
+,
+(34)
+where ˆX = RΛ−1
+s
+ˆB. Hence, we arrive at (20) and the transfer function ˆH(s) can be written as follows:
+ˆH(s) = ˆCΛ−1
+s
+ˆB
+�
+Im + RΛ−1
+s
+ˆB
+�−1
+.
+(20)
+A.2. Pole placement as in [3]
+In order to enforce both prescribed poles and certain interpolation conditions in the ROM, we follow
+the derivations from [3].
+It is to be noted that this approach is intrusive, i.e., requires access to
+the system’s matrices. Hence, a descriptor model characterized in (generalized) state-space by the
+following equations
+ΣDes :
+�
+E ˙x(t) = Ax(t) + Bu(t),
+y(t) = Cx(t),
+(35)
+with corresponding transfer function HDes(s) = C(sE − A)−1B is considered to be given. For the
+(right) interpolation points λi, i = 1, . . . , k (where interpolation is imposed), and the desired poles to
+be placed, denoted with ζj’s, the author in [3] starts by finding a row vector Cζ ∈ C1×n so that:
+Cζ
+�
+(λ1E − A)−1B · · · (λkE − A)−1B
+�
+= 01×k.
+(36)
+Preprint (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg).
+2023-01-13
+
+Q. Aumann, I. V. Gosea: Data-driven interpolation: challenges and solutions
+18
+Then, the next step is to choose projection matrices W, V ∈ Cn×k as
+WH =
+�
+��
+Cζ(ζ1E − A)−1
+...
+˜C(ζkE − A)−1
+�
+�� ,
+V =
+�(λ1E − A)−1B · · · (λkE − A)−1B�
+.
+(37)
+As explained in [3], the choice of WH above is explained by imposing the required poles for the
+reduced model, while V is chosen to match the interpolation conditions at the λi’s. Moreover, using
+these notations, it follows that ˜CV = 0. Next, put together the following matrices ˜E = WHEV,
+˜A =
+WHAV. Then, it follows that (s˜E − ˜A) loses rank when s ∈ {ζ1, . . . , ζr}. To show this, we simply
+write
+eT
+j (ζj ˜E − ˜A) = eT
+j WH(ζjE − A)V = Cζ(ζjE − A)−1(ζjE − A)V = CζV = 0.
+(38)
+Let Hζ(s) = Cζ(sE − A)−1B be a rational function in s and we note that ˆE and ˆA are a special type
+of diagonally scaled Cauchy matrices, with the following exact definition:
+˜Ei,j = −Cζ(ζiE − A)−1B − Cζ(λjE − A)−1B
+ζi − λj
+= − Hζ(ζi)
+ζi − λj
+˜Ai,j = −ζiCζ(ζiE − A)−1B − λjCζ(λjE − A)−1B
+ζi − λj
+= −ζiHζ(ζi)
+ζi − λj
+(39)
+From the definition in (39), it follows that ˜E = −D ˜BCζ,λ, where D ˜B = diag( ˜B) is a diagonal matrix.
+Similarly, it follows that ˜A = −ZD ˜BCζ,λ, where Z = diag(ζ1, . . . , ζk).
+Next, we write the other projected quantities as
+˜B = WHB =
+�Hζ(ζ1)
+· · ·
+Hζ(ζk)�T ,
+˜C = CV =
+�H(λ1)
+· · ·
+H(λk)�
+(40)
+Hence, the reduced-order linear descriptor system Σpp : (˜E, ˜A, ˜B, ˜C) matches k interpolation conditions
+and has the required poles.
+Next, we show that this model can be written equivalently in the AF format. We first note that
+ˆC = ˜C. For next step, provided that the matrix ˜E is non-singular, we remove it by incorporating
+it into the other matrices, as: ˘A = ˜E
+−1 ˜A,
+˘B = ˜E
+−1 ˜B,
+˘E = Ik,
+˘C = ˜C. We note that the two
+realizations of the interpolatory ROM, i.e., ( ˆA, ˆB, ˆC) in (21) and ( ˘A, ˘B, ˘C) introduced above, are
+actually identical. The reason for this is that ˘C = ˆC and the two ROMs match the same k moments.
+Hence, it also follows that ˘B = ˆB. Now, since ˘B = ˜E
+−1 ˜B and ˜E = −D ˜BCζ,λ, we can write that
+ˆB = −(D ˜BCζ,λ)−1 ˜B = −C−1
+ζ,λD−1
+˜B ˜B = −C−1
+ζ,λ1k.
+(41)
+Hence, the above choice of vector ˆB in (21) imposes the required poles.
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