Solving a Low-Rank Factorization Model for Matrix Completion by a Nonlinear Successive Over-Relaxation Algorithm
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Survey on Probabilistic Models of Low-Rank Matrix Factorizations
Article Survey on Probabilistic Models of Low-Rank Matrix Factorizations Jiarong Shi 1,2,*, Xiuyun Zheng 1 and Wei Yang 1 1 School of Science, Xi’an University of Architecture and Technology, Xi’an 710055, China; [email protected] (X.Z.); [email protected] (W.Y.) 2 School of Architecture, Xi’an University of Architecture and Technology, Xi’an 710055, China * Correspondence: [email protected]; Tel.: +86-29-8220-5670 Received: 12 June 2017; Accepted: 16 August 2017; Published: 19 August 2017 Abstract: Low-rank matrix factorizations such as Principal Component Analysis (PCA), Singular Value Decomposition (SVD) and Non-negative Matrix Factorization (NMF) are a large class of methods for pursuing the low-rank approximation of a given data matrix. The conventional factorization models are based on the assumption that the data matrices are contaminated stochastically by some type of noise. Thus the point estimations of low-rank components can be obtained by Maximum Likelihood (ML) estimation or Maximum a posteriori (MAP). In the past decade, a variety of probabilistic models of low-rank matrix factorizations have emerged. The most significant difference between low-rank matrix factorizations and their corresponding probabilistic models is that the latter treat the low-rank components as random variables. This paper makes a survey of the probabilistic models of low-rank matrix factorizations. Firstly, we review some probability distributions commonly-used in probabilistic models of low-rank matrix factorizations and introduce the conjugate priors of some probability distributions to simplify the Bayesian inference. Then we provide two main inference methods for probabilistic low-rank matrix factorizations, i.e., Gibbs sampling and variational Bayesian inference. -
Low-Rank Incremental Methods for Computing Dominant Singular Subspaces
Low-Rank Incremental Methods for Computing Dominant Singular Subspaces C. G. Baker∗ K. A. Gallivan∗ P. Van Dooren† Abstract This paper describes a generic low-rank incremental method for computing the dominant singular triplets of a matrix via a single pass through the matrix. This work unifies several efforts previously described in the literature. We tie the operation of the proposed method to a particular optimization- based eigensolver. This allows the description of novel methods exploiting multiple passes through the matrix. 1 Introduction Given a matrix A Rm×n,m n, the Singular Value Decomposition (SVD) of A is: ∈ ≥ Σ A = U V T , 0 where U and V are m m and n n orthogonal matrices, respectively, and Σ is a diagonal matrix whose × × elements σ1,...,σn are real, non-negative and non-increasing. The σi are the singular values of A, and the first n columns of U and V are the left and right singular vectors of A, respectively. Given k n, the singular vectors associated with the largest k singular values of A are the rank-k dominant singular≤ vectors. The subspaces associated with these vectors are the rank-k dominant singular subspaces. The components of the SVD are optimal in many respects [1], and these properties result in the occurrence of the SVD in numerous analyses, e.g., principal component analysis, proper orthogonal decomposition, the Karhunen-Loeve transform. These applications of the SVD are frequently used in problems related to, e.g., signal processing, model reduction of dynamical systems, image and face recognition, and information retrieval. -
Block Matrices in Linear Algebra
PRIMUS Problems, Resources, and Issues in Mathematics Undergraduate Studies ISSN: 1051-1970 (Print) 1935-4053 (Online) Journal homepage: https://www.tandfonline.com/loi/upri20 Block Matrices in Linear Algebra Stephan Ramon Garcia & Roger A. Horn To cite this article: Stephan Ramon Garcia & Roger A. Horn (2020) Block Matrices in Linear Algebra, PRIMUS, 30:3, 285-306, DOI: 10.1080/10511970.2019.1567214 To link to this article: https://doi.org/10.1080/10511970.2019.1567214 Accepted author version posted online: 05 Feb 2019. Published online: 13 May 2019. Submit your article to this journal Article views: 86 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=upri20 PRIMUS, 30(3): 285–306, 2020 Copyright # Taylor & Francis Group, LLC ISSN: 1051-1970 print / 1935-4053 online DOI: 10.1080/10511970.2019.1567214 Block Matrices in Linear Algebra Stephan Ramon Garcia and Roger A. Horn Abstract: Linear algebra is best done with block matrices. As evidence in sup- port of this thesis, we present numerous examples suitable for classroom presentation. Keywords: Matrix, matrix multiplication, block matrix, Kronecker product, rank, eigenvalues 1. INTRODUCTION This paper is addressed to instructors of a first course in linear algebra, who need not be specialists in the field. We aim to convince the reader that linear algebra is best done with block matrices. In particular, flexible thinking about the process of matrix multiplication can reveal concise proofs of important theorems and expose new results. Viewing linear algebra from a block-matrix perspective gives an instructor access to use- ful techniques, exercises, and examples. -
Riemannian Gradient Descent Methods for Graph-Regularized Matrix Completion
Riemannian Gradient Descent Methods for Graph-Regularized Matrix Completion Shuyu Dongy, P.-A. Absily, K. A. Gallivanz∗ yICTEAM Institute, UCLouvain, Belgium zFlorida State University, U.S.A. December 2, 2019 Abstract Low-rank matrix completion is the problem of recovering the missing entries of a data matrix by using the assumption that the true matrix admits a good low-rank approxi- mation. Much attention has been given recently to exploiting correlations between the column/row entities to improve the matrix completion quality. In this paper, we propose preconditioned gradient descent algorithms for solving the low-rank matrix completion problem with graph Laplacian-based regularizers. Experiments on synthetic data show that our approach achieves significant speedup compared to an existing method based on alternating minimization. Experimental results on real world data also show that our methods provide low-rank solutions of similar quality in comparable or less time than the state-of-the-art method. Keywords: matrix completion; graph information; Riemannian optimization; low-rank op- timization MSC: 90C90, 53B21, 15A83 1 Introduction Low-rank matrix completion arises in applications such as recommender systems, forecasting and imputation of data; see [38] for a recent survey. Given a data matrix with missing entries, the objective of matrix completion can be formulated as the minimization of an error function of a matrix variable with respect to the data matrix restricted to its revealed entries. In various applications, the data matrix either has a rank much lower than its dimensions or can be approximated by a low-rank matrix. As a consequence, restricting the rank of the matrix variable in the matrix completion objective not only corresponds to a reasonable assumption for successful recovery of the data matrix but also reduces the model complexity. -
1. Matrix Rank
L. Vandenberghe ECE133B (Spring 2020) 1. Matrix rank • subspaces, dimension, rank • QR factorization with pivoting • properties of matrix rank • low-rank matrices • pseudo-inverse 1.1 Subspace a nonempty subset V of Rm is a subspace if αx + βy 2 V for all vectors x; y 2 V and scalars α; β • all linear combinations of elements of V are in V • V is nonempty and closed under scalar multiplication and vector addition Examples • f0g, Rm • the span of a set S ⊆ Rm: all linear combinations of elements of S span¹Sº = fβ1a1 + ··· + βkak j a1;:::; ak 2 S; β1; : : : ; βk 2 Rg if S = fa1;:::; ang is a finite set, we write span¹Sº = span¹a1;:::; anº (the span of the empty set is defined as f0g) Matrix rank 1.2 Operations on subspaces three common operations on subspaces (V and W are subspaces) • intersection: V\W = fx j x 2 V; x 2 Wg • sum: V + W = fx + y j x 2 V; y 2 Wg if V\W = f0g this is called the direct sum and written as V ⊕ W • orthogonal complement: V? = fx j yT x = 0 for all y 2 Vg the result of each of the three operations is a subspace Matrix rank 1.3 Range of a matrix × T T suppose A is an m n matrix with columns a1;:::; an and rows b1;:::; bm: 2 bT 3 6 :1 7 A = a1 ··· an = 6 : 7 6 7 6 bT 7 4 m 5 Range (column space): the span of the column vectors (a subspace of Rm) range¹Aº = span¹a1;:::; anº n = fx1a1 + ··· + xnan j x 2 R g = fAx j x 2 Rng the range of AT is called the row space of A (a subspace of Rn): T range¹A º = span¹b1;:::; bmº m = fy1b1 + ··· + ymbm j y 2 R g = fAT y j y 2 Rmg Matrix rank 1.4 Nullspace of a matrix × T T suppose -
Structured Low-Rank Matrix Factorization: Global Optimality, Algorithms, and Applications
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 Structured Low-Rank Matrix Factorization: Global Optimality, Algorithms, and Applications Benjamin D. Haeffele, Rene´ Vidal, Fellow, IEEE Abstract—Recently, convex formulations of low-rank matrix factorization problems have received considerable attention in machine learning. However, such formulations often require solving for a matrix of the size of the data matrix, making it challenging to apply them to large scale datasets. Moreover, in many applications the data can display structures beyond simply being low-rank, e.g., images and videos present complex spatio-temporal structures that are largely ignored by standard low-rank methods. In this paper we study a matrix factorization technique that is suitable for large datasets and captures additional structure in the factors by using a particular form of regularization that includes well-known regularizers such as total variation and the nuclear norm as particular cases. Although the resulting optimization problem is non-convex, we show that if the size of the factors is large enough, under certain conditions, any local minimizer for the factors yields a global minimizer. A few practical algorithms are also provided to solve the matrix factorization problem, and bounds on the distance from a given approximate solution of the optimization problem to the global optimum are derived. Examples in neural calcium imaging video segmentation and hyperspectral compressed recovery show the advantages of our approach on high-dimensional -
Low-Rank Matrix Completion Yuejie Chi, Senior Member, IEEE
1 Low-Rank Matrix Completion Yuejie Chi, Senior Member, IEEE Imagine one observes a small subset of entries in a large matrix and aims to recover the entire matrix. Without a priori knowledge of the matrix, this problem is highly ill-posed. Fortunately, data matrices often exhibit low- dimensional structures that can be used effectively to regularize the solution space. The celebrated effectiveness of Principal Component Analysis (PCA) in science and engineering suggests that most variability of real-world data can be accounted for by projecting the data onto a few directions known as the principal components. Correspondingly, the data matrix can be modeled as a low-rank matrix, at least approximately. Is it possible to complete a partially observed matrix if its rank, i.e., its maximum number of linearly-independent row or column vectors, is small? Low-rank matrix completion arises in a variety of applications in recommendation systems, computer vision, and signal processing. As a motivating example, consider users’ ratings of products arranged in a rating matrix. Each rating may only be affected by a small number of factors – such as price, quality, and utility – and how they are reflected on the products’ specifications and users’ expectations. Naturally, this suggests that the rating matrix is low-rank, since the numbers of users and products are much higher than the number of factors. Often, the rating matrix is sparsely observed, and it is of great interest to predict the missing ratings to make targeted recommendations. I. RELEVANCE The theory and algorithms of low-rank matrix completion have been significantly expanded in the last decade with converging efforts from signal processing, applied mathematics, statistics, optimization and machine learning. -
Multidimensional Butterfly Factorization
JID:YACHA AID:1193 /COR [m3L; v1.215; Prn:11/05/2017; 9:36] P.1 (1-22) Appl. Comput. Harmon. Anal. ••• (••••) •••–••• Contents lists available at ScienceDirect Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha Letter to the Editor Multidimensional butterfly factorization Yingzhou Li b,∗, Haizhao Yang c, Lexing Ying a,b a Department of Mathematics, Stanford University, United States b ICME, Stanford University, United States c Department of Mathematics, Duke University, United States a r t i c l e i n f o a b s t r a c t Article history: This paper introduces the multidimensional butterfly factorization as a data-sparse Received 26 September 2015 representation of multidimensional kernel matrices that satisfy the complementary Received in revised form 30 March low-rank property. This factorization approximates such a kernel matrix of size 2017 N × N with a product of O(log N)sparse matrices, each of which contains Accepted 11 April 2017 O(N) nonzero entries. We also propose efficient algorithms for constructing this Available online xxxx Communicated by Joel A. Tropp factorization when either (i) a fast algorithm for applying the kernel matrix and its adjoint is available or (ii) every entry of the kernel matrix can be evaluated MSC: in O(1) operations. For the kernel matrices of multidimensional Fourier integral 44A55 operators, for which the complementary low-rank property is not satisfied due to a 65R10 singularity at the origin, we extend this factorization by combining it with either 65T50 a polar coordinate transformation or a multiscale decomposition of the integration domain to overcome the singularity. -
Low-Rank Matrix Completion Via Preconditioned Optimization on the Grassmann Manifold
Low-rank matrix completion via preconditioned optimization on the Grassmann manifold Nicolas Boumala, P.-A. Absilb aInria & D.I., UMR 8548, Ecole Normale Sup´erieure, Paris, France. bDepartment of Mathematical Engineering, ICTEAM Institute, Universit´ecatholique de Louvain, Belgium. Abstract We address the numerical problem of recovering large matrices of low rank when most of the entries are unknown. We exploit the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on a single Grassmann manifold. We then apply second-order Riemannian trust-region methods (RTRMC 2) and Riemannian conjugate gradient methods (RCGMC) to solve it. A preconditioner for the Hessian is introduced that helps control the conditioning of the problem and we detail preconditioned versions of Riemannian optimization algorithms. The cost of each iteration is linear in the number of known entries. The proposed methods are competitive with state-of-the-art algorithms on a wide range of problem instances. In particular, they perform well on rectangular matrices. We further note that second-order and preconditioned methods are well suited to solve badly conditioned matrix completion tasks. Keywords: Low-rank matrix completion, optimization on manifolds, Grassmann manifold, preconditioned Riemannian trust-regions, preconditioned Riemannian conjugate gradient, second-order methods, fixed-rank geometry, RTRMC, RCGMC 2000 MSC: 90C90, 53B21, 15A83 1. Introduction This paper considers the problem of recovering a low-rank m × n matrix X (m ≤ n) of which a few entries are observed, possibly with noise. Throughout, we assume that r = rank(X) is known and is much smaller than m, and we let Ω ⊂ f1 : : : mg × f1 : : : ng be the set of indices of the observed entries of X, that is, Xij is known if (i; j) is in Ω. -
Faster Matrix Completion Using Randomized SVD
Faster Matrix Completion Using Randomized SVD Xu Feng Wenjian Yu Yaohang Li BNRist, Dept. Computer Science & Tech. BNRist, Dept. Computer Science & Tech. Dept. Computer Science Tsinghua University Tsinghua University Old Dominion University Beijing, China Beijing, China Norfolk, VA 23529, USA [email protected] [email protected] [email protected] Abstract—Matrix completion is a widely used technique for handling large data set, because the singular values exceeding image inpainting and personalized recommender system, etc. a threshold and the corresponding singular vectors need to In this work, we focus on accelerating the matrix completion be computed in each iteration step. Truncated singular value using faster randomized singular value decomposition (rSVD). Firstly, two fast randomized algorithms (rSVD-PI and rSVD- decomposition (SVD), implemented with svds in Matlab or BKI) are proposed for handling sparse matrix. They make use of lansvd in PROPACK [5], is usually employed in the SVT an eigSVD procedure and several accelerating skills. Then, with algorithm [1]. Another method for matrix completion is the the rSVD-BKI algorithm and a new subspace recycling technique, inexact augmented Lagrange multiplier (IALM) algorithm [6], we accelerate the singular value thresholding (SVT) method in [1] which also involves singular value thresholding and was orig- to realize faster matrix completion. Experiments show that the proposed rSVD algorithms can be 6X faster than the basic rSVD inally proposed for the robust principal component analysis algorithm [2] while keeping same accuracy. For image inpainting (PCA) problem [7]. With artificially-generated low-rank ma- and movie-rating estimation problems (including up to 2 × 107 trices, experiments in [6] demonstrated that IALM algorithm ratings), the proposed accelerated SVT algorithm consumes 15X could be several times faster than the SVT algorithm. -
Notes APPM 5720 — P.G
Course notes APPM 5720 — P.G. Martinsson January 22, 2016 Matrix factorizations and low rank approximation The first section of the course provides a quick review of basic concepts from linear algebra that we will use frequently. Note that the pace is fast here, and assumes that you have seen these concepts in prior course-work. If not, then additional reading on the side is strongly recommended! 1. NOTATION, ETC n n 1.1. Norms. Let x = [x1; x2; : : : ; xn] denote a vector in R or C . Our default norm for vectors is the Euclidean norm 1=2 0 n 1 X 2 kxk = @ jxjj A : j=1 We will at times also use `p norms 1=p 0 n 1 X p kxkp = @ jxjj A : j=1 Let A denote an m × n matrix. For the most part, we allow A to have complex entries. We define the spectral norm of A via kAxk kAk = sup kAxk = sup : kxk=1 x6=0 kxk We define the Frobenius norm of A via 1=2 0 m n 1 X X 2 kAkF = @ jA(i; j)j A : i=1 j=1 Observe that p kAk ≤ kAkF ≤ min(m; n) kAk: 1.2. Transpose and adjoint. Given an m × n matrix A, the transpose At is the n × m matrix B with entries B(i; j) = A(j; i): The transpose is most commonly used for real matrices. It can also be used for a complex matrix, but more typically, we then use the adjoint, which is the complex conjugate of the transpose A∗ = At: 1.3. -
High-Dimensional Statistical Inference: from Vector to Matrix
University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2015 High-dimensional Statistical Inference: from Vector to Matrix Anru Zhang University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Applied Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Zhang, Anru, "High-dimensional Statistical Inference: from Vector to Matrix" (2015). Publicly Accessible Penn Dissertations. 1172. https://repository.upenn.edu/edissertations/1172 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1172 For more information, please contact [email protected]. High-dimensional Statistical Inference: from Vector to Matrix Abstract Statistical inference for sparse signals or low-rank matrices in high-dimensional settings is of significant interest in a range of contemporary applications. It has attracted significant ecentr attention in many fields including statistics, applied mathematics and electrical engineering. In this thesis, we consider several problems in including sparse signal recovery (compressed sensing under restricted isometry) and low-rank matrix recovery (matrix recovery via rank-one projections and structured matrix completion). The first part of the thesis discusses compressed sensing and affineank r minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis