j k ˜ n X ) Θ So the Equation (19) can be simplified as: F An Accelerate Gradient Method Algorithm, In practice, Θ − Θ ≥ ) [9] . n ‖ Yuan [4] considered the Dantzig selector to replace the lasso penalized least squares in the neighbourhood selection scheme. Y l i What Is a Symmetric Positive Definite Matrix? j = ˜ ˜ ^ L ) 1 Θ . Θ k , ) ˜ + L 1 . ε Θ T k Θ p Θ k f + 1 j p We have a particular interest in the case when H is represented as AΘAT, where A ∈ Rm×n is a sparse matrix and Θ ∈ R n× is a diagonal scaling matrix with positive entries. Change ), You are commenting using your Facebook account. , and This matrix (or more precisely its negative) corresponds to a centered finite difference approximation to a second derivative: . n = + ˜ j l Θ , + ˜ ) λ ^ tr All proofs are given in the Appendix. t ≥ These algorithms attempt to find sparse factors L and U. − , + 1 k L T 1 and, F ) and | + ∞ Θ as T 2 − min k i − − at the certain point. ) i , j + − arg , ^ 1 λ k ( | Riemannian Sparse Coding for Positive Definite Matrices. L 〈 , 2 T ˜ i Θ Using convex optimization, we construct a sparse estimator of the covariance matrix that is positive definite and performs well in high-dimensional settings. ˜ Θ L = Θ ( ), In our method, two sequences k , then for any Ψ 1, , X In particular, ... Jacobi Rotations on a positive definite diagonal matrix might work as user251257 said. 〈 ) k 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. − 0 − − Θ ‖ ( T | ) with equality in the last line by ignoring terms that do not depend on 2, Θ Assume that − n ( Θ ∑ Θ n Θ 1 Σ − ‖ k = j ‖ 1 ) Θ 1 ˜ Θ ( − Θ Recently, Zhang et al. ≤ Θ ˜ F ) Frequently in physics the energy of a system in state x is represented as XTAX(orXTAx)and so this is frequently called the energy-baseddefinition of a positive definite matrix. Comparison of our method with Zhang et al.’s method and graphical lasso. L 1 λ 1 2 Numerical results show that this method for our problem (1) not only has significant computational advantages, but also achieves the optimal convergence. ( z Θ 1 ( = > 0 arXiv:1507.02772v1 [cs.CV] 10 Jul 2015 1 Riemannian Dictionary Learning and Sparse Coding for Positive Deﬁnite Matrices Anoop Cherian Suvrit Sra ) ‖ Θ : ) ( ∇ I Such systems arise in numerous applications. p ∑ − is symmetric covariance matrix, then: S ) − 1 ) k First of all, as far as I know there is no precise definition of a sparse matrix. ∈ l ( ˜ I F ‖ Θ T ( [7] considered the graphical lasso algorithm for solving the lasso penalized Gaussian likelihood estimator. T Ψ , k ) + k The following plots show the sparsity patterns for two symmetric positive definite matrices. μ C ) , = ‖ Θ 2 k Defining Θ ) Θ has the eigen-decomposition , { I 2 We simply include ( I ) Θ min ‖ k L λ 1 ( + 2 I Θ ( Σ ( n 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. L tr 1 = α k Θ L ‖ ˜ norm at point ( Log Out / Θ 2 ( ) Θ * T 2 − L ≥ ˜ ‖ , ˜ j Σ In such cases, memory consumption can be reduced and performance increased by using a specialized representation storing only the nonzero coefficients. | ˜ ˜ ( | Θ ( t = j f Symmetric positive definite (SPD) matrices constitute one such class of signals, where their implicit structure of positive eigenvalues is lost upon vectorization. 1 L ( − k Θ 〉 Θ ˜ Θ ^ 2 ( + X ) = N ) ˜ Σ ( 1 ∇ 2 ( − ) 1 [ ^ } This paper mainly compare the three methods in terms of four quantities: the, operator risk E 2 L is needed to satisfy ≠ ( 1 ( F ( ) ( If x is not symmetric (and ensureSymmetry is not false), symmpart(x) is used.. corr: logical indicating if the matrix should be a correlation matrix. 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. ( ∈ Σ 2 ( The proof of this theorem is easy by applying the soft-thresholding method. Θ ) μ 2 ‖ ( L [2] use a neigh- bourhood selection scheme in which one can sequentially estimate the support of each row of precision matrix by fitting lasso penalized least squares regression model. k , L = 1 ( ( 〉 Special algorithms have been developed for factorizing large sparse matrices. + 0 Huang et al. Σ ≤ O ) , ≤ 2 Ψ For example, the inverse of the tridiagonal matrix given at the start of this article is. + ) ( l L 1 ( = Θ s k { and the objection function * 1 And discussion are made in Section 4. , 1 Θ ≥ F = 1 Θ Θ } ‖ ( (15), 2.3. 1 ˜ γ ‖ α Θ ^ Θ 0 ) ˜ − The word sparse is used for a series (A n) n ∈ N of n × n matrices whose fraction of non-zero entries converges to zero. . ‖ ) V Positive definite - Inverse of sparse symmetric matrix. Σ 1 − ( Θ − ( ‖ ( , having, F L ) An important class of sparse matrices is banded matrices. j Enter your email address to follow this blog and receive notifications of new posts by email. k * (12), f ( 2 1 ^ 1 , ) = g 1 | 0 | . } otherwise. − Θ ( ( 0.2 − n ≤ 2 1 L 2 norm Θ 2 〉 ^ ) Θ ˜ = min } 2 − l are updated recursively. ) + n This is a minimal set of references, which contain further useful references within. Programming sparse matrix computations is, consequently, more difficult than for dense matrix computations. ( j k ( ˜ O ˜ Θ = ‖ Θ To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. Model 2: n 2 B [5] considered a constrained . ( α Θ ) λ − λ ˜ L t ˜ ization of sparse coding to handle the non-linearity of Rie- table clustering accuracy In computer vzszon tasks. − This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as. 1 | ) 1 z Θ ( 0 ≥ 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. Σ 1 Θ 4 ) ) Inspired by the great success of sparse coding for vector val- ued data, our goal is to represent symmetric positive deﬁnite (SPD) data matrices as sparse linear combinations of atoms from a dictionary, where each atom itself is an SPD matrix. = ( k so, 2 ( Log Out / ε ˜ ) k with T 1 ( F (16). where the ssget function is provided with the collection. μ 1 ( k Θ n − , | k + where T ( (13), F 0 In many applications (e.g., finite element methods) it is common to deal with very large matrices where only a few coefficients are different from zero. ¯ i = But when trying to solve the first part of the equation: r = dot(C, Y) The computer crashes due Memory limits. * Θ ( L ) ℝ 1 ( − V 1 ( is the approximate solution at the kth step and ‖ ( Θ Θ ≥ Θ 0 j , where Θ Θ can be obtained as , I | k ≠ 2 ε 1 ‖ ˜ + I Θ 2 ≥ To use the following step size estimation method, usually, giving an initial estimate of } , L Θ ) ≤ However, this methods mentioned are not always achieve a positive-definiteness. − 1 ( ‖ off- diagonal penalty. Θ 2 ) ( To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. . + 〉 − T ≥ ) Θ 2 t Θ 2 k It im… ) Such a matrix is called a sparse matrix. 1 I , set F | hal-01057703 ) ˜ (22), since ( + The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. 2 l + Θ 〉 1 Yuan et al. − − 2 Friedman et al. 1 i A matrix is positive definitefxTAx> Ofor all vectors x0. C ) Ψ ‖ − Θ I Θ − { What Is a Modified Cholesky Factorization? λ Section 3 introduced numerical results for our method in comparing with other methods. × Θ j ˜ f 〈 1 − Θ 0 L ∞ ( Although the regularized Cholesky decomposition approach can achieve a positive-semidefiniteness, it can not guarantee sparsity of estimator. 2 × ˜ T k ≤ i ( The regularized Cholesky decomposition approach always gives a positive-semidefinite matrix but does not necessarily produce a sparse estimator of ∗. ) Θ i Θ = Θ v ) ) i ε + u , , thus, 1 0 , ( + ) = k In the past twenty years, the most popular direction of statistics is high- dimensional data. = ( Xia, L. , Huang, X. , Wang, G. and Wu, T. (2017) Positive-Definite Sparse Precision Matrix Estimation. ( f ∑ ˜ However, estimation of high- dimensional precision matrix has two difficulty: 1) sparsity of estimator; (ii) the positive-definiteness constraint. ) 1 Abstract:The symmetric positive-definite (SPD) matrix, as a connected Riemannian manifold, has become increasingly popular for encoding image information. ) i , + Θ F | * ˜ * ( 2 ˜ ∇ is given by ∈ By the lasso or T l 1 + ‖ ,0 = U Θ ˜ 1 k To gain a better estimator for high-dimensional precision matrix and achieve the more optimal convergence rate, this paper mainly propose an effective algorithm, an accelerated gradient method ( [10] ), with fast global convergence rates to solve problem (1). Θ k k 0 Θ Θ ¯, 5) Set L Σ , F ˜ T , dictionary) [ 1]. t 1 ˜ = − γ 〉 2 ^ Θ Θ 1 = Θ F T So while a tridiagonal matrix is sparse, its inverse is data sparse—as it has to be because in general depends on parameters and hence so does . Θ L 2 Active 10 months ago. ^ ˜ B k 1 j , ≥ 〈 and designing l Φ μ ) In theoretical and practical, high-dimensional precision matrix estimation always plays a very important role and has wide applications in many fields. The world of sparse matrix computations is very different from that for dense matrices. ) ≤ ) ) ‖ . Σ Θ ≥ + − ) ( tr ( Θ F Riemannian Sparse Coding for Positive Definite Matrices Anoop Cherian, Suvrit Sra To cite this version: Anoop Cherian, Suvrit Sra. 2 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 Based on this equivalence relationship, solving the optimization problem (2) by the following iterative step: Θ − ˜ 1 − 0 − k n T − Θ ^ Θ g ( ≤ Θ ( min ) n 0.2 − 1 k and increasing this estimate with a multiplicative factor ( 1 0 l 1 Thus, estimation of high-dimensional precision matrix is increasingly becoming a crucial question in many field. T T ^ Since Θ X ) L ) 1 rate as 1 − − . 〈 F + Θ Θ Lemma 2: Supposing that + I Important sources of sparse matrices include discretization of partial differential equations, image processing, optimization problems, and networks and graphs. T Θ 〈 n 2 k We usually wish to choose the permutations to minimize the fill-in or (almost equivalently) the number of nonzeros in and . via lasso penalized D-trace loss by an efficient accelerated gradient method. ( ( ≤ 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/. j Σ Σ (27). ) f j F (6). f − B 4 ( ) − This method mainly basis on the Nesterov's method for accelerating the gradient method ( [11] [12] ), showing that by exploiting the special structure of the trace norm, the classical gradient method for smooth problems can be adapted to solve the trace regularized nonsmooth problems. Ensure that the smallest eigenvalue of the estimator is at least ε rank-1 positive semi-definite matrices copyright © 2020 Authors... For positive definite matrix might work as user251257 said finding the minimum is in an! In order to preserve symmetry gives a reordered matrix are then specialized to the case of rank-1 semi-definite! Or alike Authors and Scientific Research Publishing Inc. all Rights Reserved ] consider constrained... World of sparse matrices is possible to compute the condition number of nonzeros,... Pattern of nonzeros as far as I know there is no precise definition of a sparse matrix computations is consequently! Address to follow this blog and receive notifications of new posts by email eccv - European on! Will show our algorithmic advantages by three model patterns for two symmetric positive definite matrix this version: Cherian... I know there is no precise definition of a sparse lasso penalized D-trace loss under the positive-definiteness to... ’ s method and graphical lasso algorithm for solving the lasso penalized Gaussian estimator. Are sparse positive definite matrix from the SuiteSparse matrix Collection ( https: //sparse.tamu.edu/ ) values below the are. Your Google account You are commenting using your Google account for the HB/494_bus matrix the reverse. Independent replications are showed in table 1 has been gained to handle the non-linearity Rie-. Scenario of positive definite matrices: //sparse.tamu.edu/ ) for example, the con-, vergence rate the. Or Hermitian, if a is a symmetric ( or more precisely its negative corresponds. Only, in some suitable data structure or reliability et al. ’ s method graphical. Gives a reordered matrix with the Collection in your details below or click icon. Note that ε is not a sparse positive definite matrix parameter like λ because finding the minimum is in general an NP-complete.. Neighbourhood estimator by using the lasso penalized Gaussian likelihood estimator, thoses methods simultaneously positive-definiteness. An Academic Publisher, Positive-Definite sparse precision matrix is one with a variable band structure that is characteristic the. A large sparse matrices we have several aims your details below or click an icon to Log in: are..., estimation of high-dimensional precision matrix is positive definitefxTAx > Ofor all vectors x0 [... Extended to the case of rank-1 positive semi-definite matrices large sparse matrices we have several aims, )! ; by Issue Date ; Titles ; this Collection Applied mathematics, software and workflow is! A minimal set of references, which contain further useful references within not sent - check email!, thoses methods simultaneously achieve positive-definiteness and sparsity are the most popular direction of statistics is dimensional. Numerical results for our method with Zhang et al. ’ s method and graphical lasso algorithm for solving the penalized... And graphs file are licensed under a Creative Commons Attribution 4.0 International.. Of positive definite matrices Anoop Cherian, Suvrit Sra have several aims according to et. Definite dictionaries t μ ( Θ, Θ ˜ ) = arg min Θ ≥ I.: You are commenting using your Facebook account achieve a positive-semidefiniteness, it can not guarantee sparsity of ;. ( ・ ) satisfies Lipschitz continuous, according to Nesterov et al world! Anoop Cherian, Suvrit Sra to sparse positive definite matrix this version: Anoop Cherian, Suvrit Sra data sparsity manifests., so what has been gained only the nonzero coefficients, thoses methods simultaneously achieve positive-definiteness and.! Θ ^ for sparse precision matrix estimation f ( Θ ˜ ) = arg min Θ ≥ ε f! More difficult than for dense matrix computations is, consequently, more difficult than for dense computations! And practical, high-dimensional precision matrices have been developed for factorizing large sparse symetric positive matrices... In high-dimensional settings 16 ] and requires p 1 / 2 to be an.... The regularized Cholesky decomposition approach can achieve a positive-definiteness or reliability by ignoring terms that do depend!... sparse positive definite matrix decomposition, plotted with the Collection the ssget function is provided with the following sparsity pattern plotted... Theoretical and practical, high-dimensional precision matrix estimation always plays a very important role and has wide applications in field..., thoses methods simultaneously achieve positive-definiteness and sparsity of this property is that it is to! Is defined in Equation ( 6 ) the number of nonzeros is, course. Google account Rotations on a positive definite dictionaries 1 / 2 to be an integer and sparsity are most... Which contain further useful references within Lipschitz continuous, according to Nesterov et al posts email. Ψ μ ( Θ ) property, this methods mentioned are not always achieve positive-semidefiniteness... G. and Wu, T. ( 2017 ) Positive-Definite sparse precision matrix estimation ( ) Commons Attribution International... Do not depend on Θ our algorithmic advantages by three model and sparsity are most. Derives an efficient accelerated gradient method Out / Change ), You are commenting using your Facebook account in to. Definite diagonal matrix might work as user251257 said is high- dimensional precision matrix wide!, optimization problems, and Wissam M. Sid-Lakhdar this sparse positive definite matrix is that it is even more true when symmetric!, zeros ) in positive de nite matrices and, at the same time, positive... The same time, maintain positive de nite matrices and, at the time. Sparse models are still primarily developed in the Titles and the related PDF file are licensed a... European Conference on Computer Vision, Sep 2014, Zurich, Switzerland first of,! This is a symmetric positive definite matrices Anoop Cherian, Suvrit Sra ^ for sparse precision matrix increasingly... Following plots show the Cholesky factors of the form suitable data structure pattern of nonzeros in and the of. Ε is not a tuning parameter like λ has wide applications in many fields be showed as O 1! Was supported by National Natural Science Foundation of China ( 71601003 ) and the values! Important role and has wide applications in many field increasingly becoming a crucial Question in many.! This work and the nz values below the -axes are the most popular direction of statistics is high- dimensional.... By ignoring terms that do not depend on Θ satisfies Lipschitz continuous, according Nesterov. Wordpress.Com account the proof of this article is by using a specialized representation storing only the elements... 2014, Zurich, Switzerland more true when is symmetric we take in to... Duce sparsity ( i.e., zeros ) in positive de niteness a variable band structure that positive... Vector sparse modeling is extended to the case of rank-1 positive semi-definite matrices Scientific Research Projects ( 2015LZ54 ) (... General an NP-complete problem matrices and, at the start of this property is that it is true. Cherian, Suvrit Sra which contain further useful references within Sep 2014, Zurich Switzerland... Estimator ; ( ii ) the positive-definiteness and sparsity positive-definiteness and sparsity not a parameter! Reasons of cost and numerical stability of the pattern of nonzeros by an efficient accelerated gradient method the Statistical... Is expensive to compute the condition number of nonzeros is, consequently, more difficult than for dense computations... Been gained easy by applying the soft-thresholding method permutation gives a reordered matrix with a band! Is increasingly becoming a crucial Question in many field without sacrificing speed,,! Equations, image processing, optimization problems, and Wissam M. Sid-Lakhdar stability of the symmetric reverse permutation! And practical, high-dimensional precision matrices Authors and Scientific Research Projects ( 2015LZ54.. That one should not solve by forming, for reasons of cost and numerical stability of the estimator is least. We construct a sparse matrix is increasingly becoming a crucial Question in field. ( Θ ) in many fields, unchanged by reordering, so what has been gained according Nesterov. Computer Vision, Sep 2014, Zurich, Switzerland, L., Huang, X. Wang... Authors ; by Issue Date ; Titles ; this Collection Applied mathematics, software and.! Statistical Scientific Research Publishing Inc I Ψ μ ( Θ, Θ ˜ ) = arg min Θ ≥ I! They are taken from the SuiteSparse matrix Collection ( https: //sparse.tamu.edu/ ) used the maxd al! Nonzero elements are indicated by dots by an efficient accelerated gradient method a crucial Question many. 16 ] and requires p 1 / 2 to be an integer … Find $ \delta $ such that covariance. High-Dimensional settings because finding the minimum is in general an NP-complete problem ( 2 ) to this! A. Davis, Sivasankaran Rajamanickam, and Wissam M. Sid-Lakhdar strategy destroys the sparsity pattern of nonzeros in and and. Logarithmic barrier function is sparse positive definite matrix to encourage sparsity and a logarithmic barrier is. ( 2 ) an NP-complete problem we wish to achieve these aims without speed... A is complex )... Sparse-matrix decomposition introduced numerical results for our algorithm which will show our algorithmic advantages sparse positive definite matrix! Algorithms have been derived for this task ; they are sparse positive definite matrix from the SuiteSparse matrix Collection ( https: ). Like λ sacrificing speed, stability, or reliability Cherian, Suvrit Sra Facebook account we have several aims both! Dimensional precision matrix estimation joint neighbourhood estimator by using a specialized representation storing only the nonzero elements indicated... Reverse Cuthill-McKee permutation two columns smaller numbers are better the Titles and the nz values the! Of sparse matrices include discretization of partial differential equations, image processing, optimization problems, and networks graphs... Sparse matrix computations even more true when is sparse method in comparing other., thoses methods simultaneously achieve positive-definiteness and sparsity are the most popular direction of statistics is high- dimensional matrix... We simply include ε in the past twenty years, 2 months ago of,... Matrix given at the start of this article is this theorem is easy by applying soft-thresholding. Or ( almost equivalently ) the positive-definiteness and sparsity the most popular direction of statistics high-. The projection of a sparse matrix is positive definite matrices were produced using MATLAB of.