Generalized Inverses and Generalized Connections with Statistics
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Solving a Low-Rank Factorization Model for Matrix Completion by a Nonlinear Successive Over-Relaxation Algorithm
SOLVING A LOW-RANK FACTORIZATION MODEL FOR MATRIX COMPLETION BY A NONLINEAR SUCCESSIVE OVER-RELAXATION ALGORITHM ZAIWEN WEN †, WOTAO YIN ‡, AND YIN ZHANG § Abstract. The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving large-scale problems, we propose a low-rank factorization model and construct a nonlinear successive over-relaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Convergence of this nonlinear SOR algorithm is analyzed. Numerical results show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclear-norm minimization algorithms. Key words. Matrix Completion, alternating minimization, nonlinear GS method, nonlinear SOR method AMS subject classifications. 65K05, 90C06, 93C41, 68Q32 1. Introduction. The problem of minimizing the rank of a matrix arises in many applications, for example, control and systems theory, model reduction and minimum order control synthesis [20], recovering shape and motion from image streams [25, 32], data mining and pattern recognitions [6] and machine learning such as latent semantic indexing, collaborative prediction and low-dimensional embedding. In this paper, we consider the Matrix Completion (MC) problem of finding a lowest-rank matrix given a subset of its entries, that is, (1.1) min rank(W ), s.t. Wij = Mij , ∀(i,j) ∈ Ω, W ∈Rm×n where rank(W ) denotes the rank of W , and Mi,j ∈ R are given for (i,j) ∈ Ω ⊂ {(i,j) : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. -
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. -
Beyond Moore-Penrose Part I: Generalized Inverses That Minimize Matrix Norms Ivan Dokmanić, Rémi Gribonval
Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix Norms Ivan Dokmanić, Rémi Gribonval To cite this version: Ivan Dokmanić, Rémi Gribonval. Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix Norms. 2017. hal-01547183v1 HAL Id: hal-01547183 https://hal.inria.fr/hal-01547183v1 Preprint submitted on 30 Jun 2017 (v1), last revised 13 Jul 2017 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix Norms Ivan Dokmani´cand R´emiGribonval Abstract This is the first paper of a two-long series in which we study linear generalized in- verses that minimize matrix norms. Such generalized inverses are famously represented by the Moore-Penrose pseudoinverse (MPP) which happens to minimize the Frobenius norm. Freeing up the degrees of freedom associated with Frobenius optimality enables us to pro- mote other interesting properties. In this Part I, we look at the basic properties of norm- minimizing generalized inverses, especially in terms of uniqueness and relation to the MPP. We first show that the MPP minimizes many norms beyond those unitarily invariant, thus further bolstering its role as a robust choice in many situations. -
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 -
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. -
Generalized Inverses of an Invertible Infinite Matrix Over a Finite
Linear Algebra and its Applications 418 (2006) 468–479 www.elsevier.com/locate/laa Generalized inverses of an invertible infinite matrix over a finite field C.R. Saranya a, K.C. Sivakumar b,∗ a Department of Mathematics, College of Engineering, Guindy, Chennai 600 025, India b Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India Received 17 August 2005; accepted 21 February 2006 Available online 21 April 2006 Submitted by M. Neumann Abstract In this paper we show that, contrary to finite matrices with entries taken from finite field, an invertible infinite matrix could have generalized inverses that are different from classical inverses. © 2006 Elsevier Inc. All rights reserved. AMS classification: Primary 15A09 Keywords: Moore–Penrose inverse; Group inverse; Matrices over finite fields; Infinite matrices 1. Introduction Let A be a finite matrix. Recall that a matrix X is called the Moore–Penrose inverse of A if it satisfies the following Penrose equations: AXA = A; XAX = X, (AX)T = AX; (XA)T = XA. Such an X will be denoted by A†. The group inverse (if it exists) of a finite square matrix A denoted by A# is the unique solution of the equations AXA = A; XAX = X; XA = AX. ∗ Corresponding author. E-mail address: [email protected] (K.C. Sivakumar). 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.02.023 C.R. Saranya, K.C. Sivakumar / Linear Algebra and its Applications 418 (2006) 468–479 469 It is well known that the group inverse of A exists iff the range space R(A) of A and the null space N(A) of A are complementary. -
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. -
Direct Methodsmethods Forfor Solvingsolving Linearlinear Systemssystems (3H.)(3H.)
NumericalNumerical MethodsMethods RafaRafałł ZdunekZdunek DirectDirect methodsmethods forfor solvingsolving linearlinear systemssystems (3h.)(3h.) ((GaussianGaussian eliminationelimination andand matrixmatrix decompositionsdecompositions)) Introduction • Solutions to linear systems • Gaussian elimination, • LU factorization, • Gauss-Jordan elimination, • Pivoting Bibliography [1] A. Bjorck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996, [2] G. Golub, C. F. Van Loan, Matrix Computations, The John Hopkins University Press, (Third Edition), 1996, [3] J. Stoer R. Bulirsch, Introduction to Numerical Analysis (Second Edition), Springer-Verlag, 1993, [4] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000, [5] Ch. Zarowski, An Introduction to Numerical Analysis for Electrical and Computer Engineers, Wiley, 2004, [6] G. Strang, Linear Algebra and Its Applications, Harcourt Brace & Company International Edition, 1998, Applications There are two basic classes of methods for solving a system of linear equations: direct and iterative methods. Direct methods theoretically give an exact solution in a (predictable) finite number of steps. Unfortunately, this does not have to be true in computational approach due to rounding errors: an error made in one step spreads in all following steps. Classical direct methods usually involve a variety of matrix factorization techniques such as the LU, LDU, LUP, Cholesky, QR, EVD, SVD, and GSVD. Direct methods are computationally efficient only for a specific class of problems where the system matrix is sparse and reveals some patterns, e.g. it is a banded matrix. Otherwise, iterative methods are usually more efficient. Solutions to linear systems m: ixrtro ma fingollow fd in thexses een beprations cqua era linem ofste syA Ax= b (1) , MN× M where =A [aij ] ∈ ℑ is a coefficient mtarix, =b []bi ∈ ℑ is a dat vector, and a N MN×( 1+ ) =x []x j ∈ ℑt ed. -
Generalized Inverse and Pseudoinverse
San José State University Math 253: Mathematical Methods for Data Visualization Lecture 6: Generalized inverse and pseudoinverse Dr. Guangliang Chen Outline Matrix generalized inverse • Pseudoinverse • Application to solving linear systems of equations • Generalized inverse and pseudoinverse Recall ... that a square matrix A ∈ Rn×n is invertible if there exists a square matrix B of the same size such that AB = BA = I In this case, B is called the matrix inverse of A and denoted as B = A−1. We already know that two equivalent ways of characterizing a square, invertible matrix A are A has full rank, i.e., rank(A) = n • A has nonzero determinant: det(A) 6= 0 • Dr. Guangliang Chen | Mathematics & Statistics, San José State University3/43 Generalized inverse and pseudoinverse Remark. For any invertible matrix A ∈ Rn×n and any vector b ∈ Rn, the linear system Ax = b has a unique solution x∗ = A−1b. MATLAB command for solving a linear system Ax = b A\b % recommended inv(A) ∗ b % avoid (especially when A is large) Dr. Guangliang Chen | Mathematics & Statistics, San José State University4/43 Generalized inverse and pseudoinverse What about general matrices? Let A ∈ Rm×n. We would like to address the following questions: Is there some kind of inverse? • Given a vector b ∈ Rm, how can we solve the linear system Ax = b? • Dr. Guangliang Chen | Mathematics & Statistics, San José State University5/43 Generalized inverse and pseudoinverse More motivation In many practical tasks such as multiple linear regression, the least squares problem arises naturally: min kAx − bk2 (where A ∈ m×n, b ∈ m are fixed) n R R x∈R If A has full column rank (i.e., rank(A) = n ≤ m), then the above problem has a unique solution x∗ = (AT A)−1AT b We want to better understand the matrices: (AT A)−1AT (pseudoinverse): Optimal solution is x∗ = (AT A)−1AT b; • A(AT A)−1AT (projection matrix): Closest approximation of b is Ax∗ = • A(AT A)−1AT b Dr. -
T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product
T-Jordan Canonical Form and T-Drazin Inverse based on the T-Product Yun Miao∗ Liqun Qiy Yimin Weiz October 24, 2019 Abstract In this paper, we investigate the tensor similar relationship and propose the T-Jordan canonical form and its properties. The concept of T-minimal polynomial and T-characteristic polynomial are proposed. As a special case, we present properties when two tensors commutes based on the tensor T-product. We prove that the Cayley-Hamilton theorem also holds for tensor cases. Then we focus on the tensor decompositions: T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained. When a F-square tensor is not invertible with the T-product, we study the T-group inverse and T-Drazin inverse which can be viewed as the extension of matrix cases. The expression of T-group and T-Drazin inverse are given by the T-Jordan canonical form. The polynomial form of T-Drazin inverse is also proposed. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverse can be given by a limiting formula. Keywords. T-Jordan canonical form, T-function, T-index, tensor decomposition, T-Drazin inverse, T-group inverse, T-core-nilpotent decomposition. AMS Subject Classifications. 15A48, 15A69, 65F10, 65H10, 65N22. arXiv:1902.07024v2 [math.NA] 22 Oct 2019 ∗E-mail: [email protected]. School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. of China. Y. Miao is supported by the National Natural Science Foundation of China under grant 11771099.