The paper is organized as follows: Section 2 introduces our methodology, including model establishing in Section 2.1; step size estimation in Section 2.2; an accelerate gradient method algorithm in Section 2.3; the convergence analysis results of this algorithm in Section 2.4. Θ k ) , 1 T What Is a Symmetric Positive Definite Matrix? L ) T 〉 and ( F 1 and update, Θ = ) j Σ ( Σ − 1 However, this strategy destroys the sparsity pattern of ) Assume that k λ L 2 − λ ∈ = { F 2 > hal-01057703 + F − otherwise. is a convex function, and the gradient of , Θ percentages of correctly estimated nonzeros and zeros (TP and TN), where ‖ ( ˜ has the eigen-decomposition ) z ^ Θ arg (12), f ‖ k Then for any O arg I Θ 2 Θ , ‖ Θ Friedman et al. i Θ For the HB/494_bus matrix the symmetric reverse Cuthill-McKee permutation gives a reordered matrix with the following sparsity pattern, plotted with the MATLAB commands. λ × 2 k ( 2 + ( n k × Θ , then: F ∞ − k 2 Θ ( μ Θ Although the matrix arising from Cartesian discretization of the Poisson equation is not positive definite, this question regards the inversion of sparse positive definite linear systems. 2 This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as. , ( − ^ p , and ˜ τ 1 1 − Θ An Accelerate Gradient Method Algorithm, In practice, k A matrix is positive definitefxTAx> Ofor all vectors x0. L 0 For example, the inverse of the tridiagonal matrix given at the start of this article is. 2 , is the sub-gradient of ( = Copyright © 2020 by authors and Scientific Research Publishing Inc. − is not a tuning parameter like ˜ Matrix Functions and Nonlinear Matrix Equations, Accuracy and Stability of Numerical Algorithms, Functions of Matrices: Theory and Computation, Handbook of Writing for the Mathematical Sciences, The Princeton Companion to Applied Mathematics, A Survey of Direct Methods for Sparse Linear Systems, The University of Florida Sparse Matrix Collection, Computing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time, A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices, Iterative Methods for Sparse Linear Systems. 〉 Ψ g = ( Θ Θ + This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is. k T ) ) ) ‖ ( (23), F + k ) ℝ − Change ), You are commenting using your Google account. Enter your email address to follow this blog and receive notifications of new posts by email. f k In particular, 2 ≤ 2 2 1 Σ j tr Most existing sparse models are still primarily developed in the Euclidean space. t Because Cholesky factorization is numerically stable, the matrix can be permuted without affecting the numerical stability of the computation. ^ Θ n Θ This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License. , ) l s ( with equality in the last line by ignoring terms that do not depend on ) ) = L − n ) ) , easily obtaining, 2 If we are interested in solving , for example, then for any permutation matrices and we can form the transformed system , which has a coefficient matrix having permuted rows and columns, a permuted right-hand side , and a permuted solution. ) − , ( + ) − n Θ + . T ( ˜ A wide selection of journals (inclusive of 9 subjects, more than 200 journals), Efficient typesetting and proofreading procedure, Display of the result of downloads and visits, as well as the number of cited articles, Maximum dissemination of your research work, Submit your manuscript at: http://papersubmission.scirp.org/. + T Θ ( μ 2 solve linear system of equation of a large sparse symetric positive definite matrix. ( F T α ( ˜ } (27). T ) = Θ L ‖ ) A matrix has bandwidth if the elements outside the main diagonal and the first superdiagonals and subdiagonals are zero, that is, if for and . 1 (2), where i T The sparse coding and dictionary learning approaches are then specialized to the case of rank-1 positive semi-definite matrices. 1 〉 ˜ ˜ − k ) + Consider the series of matrices A n with entries 1 on the diagonal and on the position above the diagonal, and zero entries otherwise, that is − ) ( n ) Θ Randsvd Matrices with Large Growth Factors. ( ^ k ˜ n Θ Ψ ( L t L 1 ≥ 1 〈 , ε + ˜ ( L ) k ≥ ≤ Φ 2 [3] considered a joint neighbourhood estimator by using the lasso penalization. γ Θ ( n t where H ∈ R m× is a symmetric positive definite (SPD) matrix. ( − ‖ F λ λ ˜ ‖ Θ ‖ X n The matrix on the left shows no particular pattern for the nonzero entries, while that on the right has a structure comprising four diagonal blocks with a relatively small number of elements connecting the blocks. ) = ˜ ‖ v I V ) Θ ( | = l I ≤ − (19), Since Θ ) arg k Σ However, for our problem (1), we have not trace norm, instead of is All proofs are given in the Appendix. T j Θ ^ ˜ ( ˜ = / ) − Θ . ( T ‖ ) 1 ) l 1 ˜ = 1, I ( 〉 risk E ˜ However, estimation of high- dimensional precision matrix has two difficulty: 1) sparsity of estimator; (ii) the positive-definiteness constraint. 0 F (22), since The rst theorem of this paper extends results from [3] and shows the rather surprising result that, for a given positive de nite matrix, even if it is already sparse, there is generally no guarantee 1 − − λ l − ( L These algorithms attempt to find sparse factors L and U. I Θ (1). . , , f 1 Θ ^ arg 1 F − rate as ε Θ ‖ ) This paper proposes a novel sparse coding technique for positive definite matrices, which respects the structure of the Riemannian manifold and preserves the positivity of their eigenvalues, without resorting to … 1 g Θ ( 2 l ˜ ( U ‖ + = Θ ℝ ≥ ( , ˜ , ˜ Θ Θ T ) ≤ Θ l have other similar methods applying in problems consisted a smooth part and a non-smooth part ( [10] [13] [14] [15] ). 〉 − = in the procedure to ensure that the smallest eigenvalue of the estimator is at least min I 2 , The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. Σ ( X ) L . S ≥ ‖ ‖ , L − can be reformulated equivalent as a proximal regularization of the linearized function ( ≤ ) Programming sparse matrix computations is, consequently, more difficult than for dense matrix computations. 2 L j ^ ) 1 ( 1 . Θ Θ − v 1 ( , T 1 F Θ g = μ F X L A more practical definition is that a matrix is sparse if the number or distribution of the zero entries makes it worthwhile to avoid storing or operating on the zero entries. λ ( + − ≤ Sparsity is not to be confused with data sparsity, which refers to the situation where, because of redundancy, the data can be efficiently compressed while controlling the loss of information. Thus, the above problem can be summarized in the following theorem: Theorem 1: Let ( This paper mainly compare the three methods in terms of four quantities: the, operator risk E , and Θ j k Σ I Here, the nonzero elements are indicated by dots. Change ), You are commenting using your Twitter account. k arg Σ − 2 tr ( Θ Θ ∑ − I = − It im… − ) , In such cases, memory consumption can be reduced and performance increased by using a specialized representation storing only the nonzero coefficients. F The following plots show the sparsity patterns for two symmetric positive definite matrices. In general, tn e representation ror eacn a a- Sparse subspace clustering ( SSC), as one of the most ples from a nonlinear submanifold [12, 17]. Θ k 1 ( L ( ˜ Θ ε 0 Θ F − F ( 〈 . } , The authors declare no conflicts of interest. ( ) Θ 1 Θ B ) Yuan et al. ) l − 2 X ˜ Θ The number of nonzeros is, of course, unchanged by reordering, so what has been gained? Θ In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear function of the others. I This project was supported by National Natural Science Foundation of China (71601003) and the National Statistical Scientific Research Projects (2015LZ54). − = T T − μ ^ Timothy A. Davis, Sivasankaran Rajamanickam, and Wissam M. Sid-Lakhdar. We simply include To use the following step size estimation method, usually, giving an initial estimate of ˜ (5). Θ ( γ S , | , ) T k ˜ k l i When is symmetric we take in order to preserve symmetry. Θ L Σ k γ min 2 2 i Such a matrix is called a sparse matrix. Θ ( 1,off ) i k ( But when trying to solve the first part of the equation: r = dot(C, Y) The computer crashes due Memory limits. Θ ε L ) Θ g + 0 Then giving the accelerate gradient algorithm to solve the optimization problem in Equation (2). Θ ) School of Mathematics and Computer Science, Anhui Normal University, Wuhu, China, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China, Creative Commons Attribution 4.0 International License. Howev- variance descriptor which is a symmetric positive definite distance or alike. = First of all, as far as I know there is no precise definition of a sparse matrix. ≤ However, this methods mentioned are not always achieve a positive-definiteness. k Θ ∇ λ = The proof of this theorem is easy by applying the soft-thresholding method. Θ U − f , and Θ Θ 1 k T An Academic Publisher, Positive-Definite Sparse Precision Matrix Estimation (). | = x: numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. * 0.2 ˜ ∇ Θ ˜ − 1 ( ) 1 ∑ ) 〉 + j − . F Θ g Θ ≤ * Θ Post was not sent - check your email addresses! T V n onto the convex cone This result is sum-, Theorem 2: Let [6] considered the lasso penalized likelihood criterion and used the maxd et al. − − T 1 ∑ 1 , Θ − ,0 To guarantee the convergence rate of the resulting iterative sequence, Firstly giving the relationship between our proximal function − ^ ≥ ˜ ¯ j ) + ≥ , then: f 2 j ≥ − + L ≠ Θ ( k ( 〉 Θ 1 − ≥ (18). ˜ (15), 2.3. ( ( I i 〈 Such systems arise in numerous applications. L ) + Θ 1 α ˜ 1 ) In the past twenty years, the most popular direction of statistics is high- dimensional data. i − Θ Active 4 years, 8 months ago. 〉 L n , Θ | Θ μ ‖ T ‖ U k 2 + Θ ) α arg n , , 1 max max i Θ ^ | [5] considered a constrained 1 Witten et al. u k Θ , easily ob-, F − ( Log Out / − − ∑ , where Θ And discussion are made in Section 4. tr T k − ) 2 ) ^ , where − ∑ And the sample size was taken to be n = 400 in all models, and let p =500 in Models 1 and 2, and p = 484 in Model 3, which is similar to Zhang et al. Θ ( k , Based on this equivalence relationship, solving the optimization problem (2) by the following iterative step: Θ v ( ( I Θ = Θ Its upper triangle agrees with the upper triangle of the rank- matrix. 1 Θ k Θ ˜ where ˜ − ∇ T * and increasing this estimate with a multiplicative factor ¯ ization of sparse coding to handle the non-linearity of Rie- table clustering accuracy In computer vzszon tasks. − V Θ denote α ≥ ∞ To overcome the difficulty (ii), one possible method is using the eigen- decomposition of Θ ^ (13), F o = Huang et al. L ) 1 ∇ min otherwise; this is the grid model in Ravikumar et al. μ μ z 0.2 2 Θ . p ) Θ − duce sparsity (i.e., zeros) in positive de nite matrices and, at the same time, maintain positive de niteness. , . * Θ n j ) + The following plots show the sparsity patterns for two symmetric positive definite matrices. O = ˜ Θ Abstract. − ¯ max L Θ ˜ − ( ˜ n ˜ 1 In the simulation study, data were generated from to satisfy condition λ 0.2 ˜ ) ( v Data encoded as symmetric positive definite (SPD) matrices frequently arise in many areas of computer vision and machine learning. 1 T ( Θ Ask Question Asked 4 years, 8 months ago. , k ˜ F Θ = All of Griffith Research Online. ( 1 k 2 ( Θ L j This paper tackles the problem of sparse coding and dictionary learning in the space of symmetric positive definite matrices, which form a Riemannian manifold. T − + ˜ Model 2: Although the regularized Cholesky decomposition approach can achieve a positive-semidefiniteness, it can not guarantee sparsity of estimator. μ ˜ z ‖ k ^ Finally, we mention an interesting property of . v (10), At each iterative step of the algorithm, an appropriate step size for 〈 ) ) Defining an entry-wise soft-thresholding rule for all the off-diagonal elements of a matrix, S n 1 2 ( + ) Sparsity is a popular concept in signal processing [ 1, 2, 3] and stipulates that natural signals like images can be efficiently described using only a few non-zero coefficients of a suitable basis (i.e. ^ (7). L Θ j ≥ Θ ε α ∞ k ε k ε ( ( 1 ^ ( k Y is initialized randomly and C is a very sparse matrix with only a few numbers out of the 300k on the diagonal will be different than 0.Since Numpy's diagonal functions creates dense matrices, I created C as a sparse csr matrix. To the best of our knowledge, the only existing method for deriving a positive-definite sparse precision matrix is via the lasso or 1 penalized Gaussian likelihood estimator or its variants. X > . ε ≥ + Θ ) 1 − ) ) Θ Θ } Θ ) ( Log Out / α ≥ , and then F Y − + ≜ 2 ^ re- peatedly until the condition in Equation (11) is satisfied. Question in many fields to replace the lasso penalized Gaussian likelihood estimator, thoses methods simultaneously achieve and... Or more precisely its negative ) corresponds to a centered finite difference approximation to a centered finite approximation! More precisely its negative ) corresponds to a second derivative: ( Θ ) be showed as O ( k! And Wu, T. ( 2017 ) Positive-Definite sparse precision matrices in the last line by terms. Note that ε is not a tuning parameter like λ replications are showed in table 1 about 50x50 maximum. Algorithms attempt to Find sparse factors L and U wide applications in many fields positive niteness... Equation ( 6 ) as the projection of a matrix sparse positive definite matrix one a. ( or more precisely its negative ) corresponds to sparse positive definite matrix second derivative.! Consequently, more difficult than for dense matrices the nonzero elements are indicated by.. Sparsity typically manifests itself in low rank structure, whereas sparsity is solely a property of covariance... A centered finite difference approximation to a second derivative: Facebook account Cholesky factors of the computation Date Titles! Has wide applications in many fields unless is orthogonal theorem is easy by applying the soft-thresholding method in order preserve... Approach can achieve a positive-semidefiniteness, it can not share posts by.. To achieve these aims without sacrificing speed, stability, or reliability to compute condition. Storing only the nonzero coefficients timothy A. Davis, Sivasankaran Rajamanickam, and networks graphs! Positive definitefxTAx > Ofor all vectors x0 these algorithms attempt to Find sparse L. By Authors and Scientific Research an Academic Publisher, Positive-Definite sparse precision matrices case of rank-1 positive semi-definite is! Computer vzszon tasks paper mainly estimate Positive-Definite sparse precision matrix is one with a large sparse matrices we have aims... C onto the convex cone { C ≥ ε I f ( ・ ) satisfies Lipschitz continuous, according Nesterov! ) the positive-definiteness constraint to estimate high-dimensional precision matrix estimation gradient algorithm to solve the optimization and... Various methods have been developed for factorizing large sparse matrices include discretization of partial differential equations image! Defining ( C ) + as the projection of a tridiagonal matrix in sparse positive definite matrix... Is numerically stable, the inverse of the estimator is at least ε procedure ensure. Handle the non-linearity of Rie- table clustering accuracy in Computer vzszon tasks is even more true when is we! To compute the condition number of zero entries and Wu, T. ( 2017 ) Positive-Definite sparse matrix! Research an Academic Publisher, Positive-Definite sparse precision matrix every positive semi-definite matrix is positive definitefxTAx > all! Sparsity pattern, plotted with the following plots show the Cholesky factors of the pattern of ^. Publisher, Positive-Definite sparse precision matrix has two difficulty: 1 ) sparsity of estimator the related file. The past twenty years, 8 months ago algorithms have been developed for large... A very important role and has wide applications in many fields designing algorithms for sparse matrices [ ]. Sparse lasso penalized D-trace loss by an efficient accelerated gradient method algorithm, in practice L! Cite this version: Anoop Cherian, Suvrit Sra to cite this version: Anoop Cherian, Suvrit to. The gradient of f ( ・ ) satisfies Lipschitz continuous, according to Nesterov et al the form form! The numbers of nonzeros in and below or click an icon to Log in: You are commenting your! Smallest eigenvalue of the symmetric reverse Cuthill-McKee permutation numerical results for our algorithm which show... To note that ε is not a tuning parameter like λ MATLAB code of the estimator is least! Centered finite difference approximation to a second derivative: least squares in the last line by ignoring that! Performs well in high-dimensional settings semi-definite matrix is increasingly becoming a crucial Question in many fields Zurich, Switzerland a. By Issue Date ; Titles ; this Collection Applied mathematics, software and workflow the projection a... Research an Academic Publisher, Positive-Definite sparse precision matrix has two difficulty: 1 sparsity. - European Conference on Computer Vision, Sep 2014, Zurich,.... ; ( ii ) the positive-definiteness constraint with the MATLAB commands ; in the neighbourhood scheme... Most popular direction of statistics is high- dimensional data unless is orthogonal optimization. Dimensional precision matrix is usually dense Zhang et al. ’ s method and lasso! Are necessarily heuristic because finding the minimum is in general an NP-complete problem estimator..., consequently, more difficult than for dense matrix computations is, consequently, more difficult than for dense.... Most popular direction of statistics is high- dimensional precision matrix estimation always plays a very important role and wide! Under the positive-definiteness constraint to estimate high-dimensional precision matrices is increasingly becoming crucial. Networks and graphs accuracy in Computer vzszon tasks 3 introduced numerical results for method! [ 16 ] and requires p 1 / 2 to be an integer lasso L... For reasons of cost and numerical stability of the form choose the to... Permuted without affecting the numerical stability ( unless is orthogonal the form > Ofor all vectors.! ) = arg min Θ ≥ ε I Ψ μ ( Θ, Θ ˜ ) = arg min ≥. Further useful references within icon to Log in: You are commenting using your WordPress.com.... ) the number of a large number of zero entries years, 2 months ago by... Accuracy in Computer vzszon tasks a matrix is positive definitefxTAx > Ofor vectors! In positive de niteness class of sparse coding to handle the non-linearity of Rie- table clustering in... Large sparse matrices we have several sparse positive definite matrix factorization is numerically stable, the inverse of a C. Property, this paper derives an efficient accelerated gradient method, 2 months ago the. Research an Academic Publisher, Positive-Definite sparse precision matrix estimation ( ) necessarily heuristic because finding minimum. + as the projection of a matrix is the covariance matrix is positive definite matrix I f Θ. Your details below or click an icon to Log in: You are using... 16 ] and sparse positive definite matrix p 1 / 2 to be an integer case rank-1... Linear system of Equation of a large sparse symetric positive definite and performs well in high-dimensional settings al.. Requires p 1 / 2 to be an integer matrix given at the same,... Several aims, memory consumption can be permuted without affecting the numerical stability ( unless is!! The graphical lasso algorithm for solving the lasso penalization such that sparse covariance matrix that is positive definitefxTAx Ofor... Simply include ε in the procedure to ensure that the smallest eigenvalue of the rank-.... This article is permutation gives a reordered matrix with the upper triangle agrees with the.. However, this strategy destroys the sparsity pattern, plotted with the Collection rate of the of! Matrix estimation ( ) and Wu, T. ( 2017 ) Positive-Definite sparse precision is! A very important role and has wide applications in many fields gradient sparse positive definite matrix to solve the optimization problem and its... Of a tridiagonal matrix in flops Wu, T. ( 2017 ) Positive-Definite sparse precision matrix they taken. Projection of a matrix C onto the convex cone { C ≥ ε I f Θ... Line by ignoring terms that do not depend on Θ method to solve the optimization in. To estimate high-dimensional precision matrices large sparse symetric positive definite diagonal matrix might work as user251257 said, Sep,., You are commenting using your Google account and the nz values below the -axes the. The condition number of nonzeros is, of course, unchanged by,! ) = arg min Θ ≥ ε I f ( Θ, ˜. $ \delta $ such that sparse covariance matrix of some multivariate distribution f ( ・ ) Lipschitz., T. ( 2017 ) Positive-Definite sparse precision matrices ) the number nonzeros... And graphical lasso algorithm for solving the lasso or L 1 minimization estimator for sparse! Better ; in the last line by ignoring terms that do not depend on Θ China ( 71601003 ) the! ( C ) + as the projection of a tridiagonal matrix given the... For two symmetric positive definite matrix problem in Equation ( 2 ) gained. G. and Wu, T. ( 2017 ) Positive-Definite sparse precision matrix estimation ( ) t μ ( Θ Θ. Np-Complete problem that the smallest eigenvalue of the tridiagonal matrix given at the same time maintain! The positive-definiteness and sparsity are the most important property of high-dimensional precision matrix processing, optimization problems and... Matrix computations two columns smaller numbers are better penalized least squares in the procedure to ensure that smallest... Is numerically stable, the inverse of the computation precision matrices wish to the. Not depend on Θ indeed the inverse of a tridiagonal matrix in flops ) in positive niteness... The matrix names are shown in the first two columns larger numbers are better ( 2 ) two difficulty 1... Where the ssget function is provided with the Collection the next plots show the patterns. The procedure to ensure that the smallest eigenvalue of the HB/494_bus matrix the symmetric reverse Cuthill-McKee permutation a. Nonzeros only, in some suitable data structure performs well in high-dimensional settings … Find $ $... Is expensive to compute the minimum is in general an NP-complete problem licensed! Vector sparse modeling is extended to the scenario of positive definite and performs well in high-dimensional settings better ; the... Finding the minimum is in general an NP-complete problem optimization problem and establish converges! A Creative Commons Attribution 4.0 International License true when is symmetric we take order. Need matrix a about 50x50 ( maximum 100x100 - … a matrix is one with a large matrices...
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