A Momentum Accelerated Algorithm for ReLU-based Nonlinear Matrix Decomposition
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
Recently, there has been a growing interest in the exploration of Nonlinear Matrix Decomposition (NMD) due to its close ties with neural networks. NMD aims to find a low-rank matrix from a sparse nonnegative matrix with a per-element nonlinear function. A typical choice is the Rectified Linear Unit (ReLU) activation function. To address over-fitting in the existing ReLU-based NMD model (ReLU-NMD), we propose a Tikhonov regularized ReLU-NMD model, referred to as ReLU-NMD-T. Subsequently, we introduce a momentum accelerated algorithm for handling the ReLU-NMD-T model. A distinctive feature, setting our work apart from most existing studies, is the incorporation of both positive and negative momentum parameters in our algorithm. Our numerical experiments on real-world datasets show the effectiveness of the proposed model and algorithm. Moreover, the code is available at https://github.com/nothing2wang/NMD-TM.
An Accelerated Alternating Partial Bregman Algorithm for ReLU-based Matrix Decomposition
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
We argue the Fisher information matrix (FIM) of one hidden layer networks with the ReLU activation function. For a network, let $W$ denote the $d \times p$ weight matrix from the $d$-dimensional input to the hidden layer consisting of $p$ neurons, and $v$ the $p$-dimensional weight vector from the hidden layer to the scalar output. We focus on the FIM of $v$, which we denote as $I$. Under certain conditions, we characterize the first three clusters of eigenvalues and eigenvectors of the FIM. Specifically, we show that 1) Since $I$ is non-negative owing to the ReLU, the first eigenvalue is the …
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Peking University（北京大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
This paper focuses on establishing $L^2$ approximation properties for deep ReLU convolutional neural networks (CNNs) in two-dimensional space. The analysis is based on a decomposition theorem for convolutional kernels with a large spatial size and multi-channels. Given the decomposition result, the property of the ReLU activation function, and a specific structure for channels, a universal approximation theorem of deep ReLU CNNs with classic structure is obtained by showing its connection with one-hidden-layer ReLU neural networks (NNs). Furthermore, approximation properties are obtained for o…
Topological Signatures of ReLU Neural Network Activation Patterns
Applied Math and Computational Sciences University of Pennsylvania（应用数学与计算科学大学公园大学）；Department of Mathematics Colorado State University（数学系科罗拉多州立大学）；Departments of Mechanical Engineering & Computational Mathematics, Sciences and Engineering (CMSE) Michigan State University（机械工程与计算数学、科学与工程系密歇根州立大学）；Department of Mathematics University of Washington（数学系华盛顿大学）；Georgia Tech Research Institute（佐治亚理工研究学院）
Poisson Hyperplane Processes with Rectified Linear Units
Institute of Mathematical Sciences, ShanghaiTech University, China（上海科技大学数学科学研究所）；Department of Statistics and Actuarial Science, Simon Fraser University, Canada（西蒙弗雷泽大学统计与精算科学系）
We present a framework to derive upper bounds on the number of regions that feed-forward neural networks with ReLU activation functions are affine linear on. It is based on an inductive analysis that keeps track of the number of such regions per dimensionality of their images within the layers. More precisely, the information about the number regions per dimensionality is pushed through the layers starting with one region of the input dimension of the neural network and using a recursion based on an analysis of how many regions per output dimensionality a subsequent layer with a certain width …
Approximating Simple ReLU Networks based on Spectral Decomposition of Fisher Information
Joint Graduate School of Mathematics for Innovation（创新数学联合研究生学校）；Kyushu University（九州大学）；Faculty of Information Science and Electrical Engineering（信息科学与电气工程学系）
A ReLU neural network determines/is a continuous piecewise linear map from an input space to an output space. The weights in the neural network determine a decomposition of the input space into convex polytopes and on each of these polytopes the network can be described by a single affine mapping. The structure of the decomposition, together with the affine map attached to each polytope, can be analyzed to investigate the behavior of the associated neural network.
Previous work has shown that a neural network with the rectified linear unit (ReLU) activation function leads to a convex polyhedral decomposition of the input space. These decompositions can be represented by a dual graph with vertices corresponding to polyhedra and edges corresponding to polyhedra sharing a facet, which is a subgraph of a Hamming graph. This paper illustrates how one can utilize the dual graph to detect and analyze adversarial attacks in the context of digital images. When an image passes through a network containing ReLU nodes, the firing or non-firing at a node can be enco…
This paper presents a new reachability analysis approach to compute interval over-approximations of the output set of feedforward neural networks with input uncertainty. We adapt to neural networks an existing mixed-monotonicity method for the reachability analysis of dynamical systems and apply it to each partial network within the main network. This ensures that the intersection of the obtained results is the tightest interval over-approximation of the output of each layer that can be obtained using mixed-monotonicity on any partial network decomposition. Unlike other tools in the literature…
We consider the problem of structured canonical polyadic decomposition. If the size of the problem is very big, then stochastic gradient approaches are viable alternatives to classical methods, such as Alternating Optimization and All-At-Once optimization. We extend a recent stochastic gradient approach by employing an acceleration step (Nesterov momentum) in each iteration. We compare our approach with state-of-the-art alternatives, using both synthetic and real-world data, and find it to be very competitive.
Incorporating a so-called "momentum" dynamic in gradient descent methods is widely used in neural net training as it has been broadly observed that, at least empirically, it often leads to significantly faster convergence. At the same time, there are very few theoretical guarantees in the literature to explain this apparent acceleration effect. Even for the classical strongly convex quadratic problems, several existing results only show Polyak's momentum has an accelerated linear rate asymptotically. In this paper, we first revisit the quadratic problems and show a non-asymptotic accelerated l…
Monash University（莫纳什大学）；University of Queensland（昆士兰大学）；University of Waterloo（滑铁卢大学）
We present Nesterov-type acceleration techniques for Alternating Least Squares (ALS) methods applied to canonical tensor decomposition. While Nesterov acceleration turns gradient descent into an optimal first-order method for convex problems by adding a momentum term with a specific weight sequence, a direct application of this method and weight sequence to ALS results in erratic convergence behaviour. This is so because the tensor decomposition problem is non-convex and ALS is accelerated instead of gradient descent. Instead, we consider various restart mechanisms and suitable choices of mome…
Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation
School of Mathematics and Statistics, Southwest University（数学与统计学学院，西南大学）；Department of Statistics and Data Science, The Chinese University of Hong Kong（统计与数据科学系，香港中文大学）；School of Mathematics and Statistics, Northwestern Polytechnical University（数学与统计学学院，西北工业大学）；Faculty of Computer Science and Control Engineering, Shenzhen University of Advanced Technology（计算机科学与控制工程学院，深圳先进技术大学）
Support vector machine (SVM) training is an active research area since the dawn of the method. In recent years there has been increasing interest in specialized solvers for the important case of linear models. The algorithm presented by Hsieh et al., probably best known under the name of the "liblinear" implementation, marks a major breakthrough. The method is analog to established dual decomposition algorithms for training of non-linear SVMs, but with greatly reduced computational complexity per update step. This comes at the cost of not keeping track of the gradient of the objective any more…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）；Seoul National University College of Medicine（首尔大学医学院）；Seoul National University Bundang Hospital（首尔大学盆唐医院）
The annihilating filter-based low-rank Hankel matrix approach (ALOHA) is one of the state-of-the-art compressed sensing approaches that directly interpolates the missing k-space data using low-rank Hankel matrix completion. The success of ALOHA is due to the concise signal representation in the k-space domain thanks to the duality between structured low-rankness in the k-space domain and the image domain sparsity. Inspired by the recent mathematical discovery that links convolutional neural networks to Hankel matrix decomposition using data-driven framelet basis, here we propose a fully data-d…
University of California, Los Angeles（加州大学洛杉矶分校）；University of Texas at Arlington（德克萨斯大学阿灵顿分校）；Sun Yat-sen University（中山大学）
Robust principal component analysis (RPCA) is a widely used tool for dimension reduction. In this work, we propose a novel non-convex algorithm, coined Iterated Robust CUR (IRCUR), for solving RPCA problems, which dramatically improves the computational efficiency in comparison with the existing algorithms. IRCUR achieves this acceleration by employing CUR decomposition when updating the low rank component, which allows us to obtain an accurate low rank approximation via only three small submatrices. Consequently, IRCUR is able to process only the small submatrices and avoid expensive computin…
University of Texas at Arlington（德克萨斯大学阿灵顿分校）；University of California, Los Angeles（加州大学洛杉矶分校）
Robust Principal Component Analysis (PCA) has received massive attention in recent years. It aims to recover a low-rank matrix and a sparse matrix from their sum. This paper proposes a novel nonconvex Robust PCA algorithm, coined Riemannian CUR (RieCUR), which utilizes the ideas of Riemannian optimization and robust CUR decompositions. This algorithm has the same computational complexity as Iterated Robust CUR, which is currently state-of-the-art, but is more robust to outliers. RieCUR is also able to tolerate a significant amount of outliers, and is comparable to Accelerated Alternating Proje…
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Center for Excellence in Animal Evolution and Genetics, Chinese Academy of Sciences（中国科学院动物进化与遗传前沿交叉卓越创新中心）；Yunnan University（云南大学）
Matrix decomposition is one of the fundamental tools to discover knowledge from big data generated by modern applications. However, it is still inefficient or infeasible to process very big data using such a method in a single machine. Moreover, big data are often distributedly collected and stored on different machines. Thus, such data generally bear strong heterogeneous noise. It is essential and useful to develop distributed matrix decomposition for big data analytics. Such a method should scale up well, model the heterogeneous noise, and address the communication issue in a distributed sys…
