SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 112-133, March 2024. Abstract. The problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the [math] largest eigenvalues of the normalized adjacency matrix (equivalently, the [math] smallest eigenvalues of the normalized graph Laplacian) are computed. It is shown that the information necessary for partitioning is contained in the subspace spanned by the [math] eigenvectors. The partitioning is encoded in a matrix [math] in indicator form, which is computed by approximating the eigenvector matrix by a product of [math] and an orthogonal matrix. A measure of the distance of a graph to being [math]-partitionable is defined, as well as two cut (cost) functions, for which Cheeger inequalities are proved; thus the relation between the eigenvalue and partitioning problems is established. Numerical examples are given that demonstrate that the partitioning algorithm is efficient and robust.
{"title":"Multiway Spectral Graph Partitioning: Cut Functions, Cheeger Inequalities, and a Simple Algorithm","authors":"Lars Eldén","doi":"10.1137/23m1551936","DOIUrl":"https://doi.org/10.1137/23m1551936","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 112-133, March 2024. <br/> Abstract. The problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the [math] largest eigenvalues of the normalized adjacency matrix (equivalently, the [math] smallest eigenvalues of the normalized graph Laplacian) are computed. It is shown that the information necessary for partitioning is contained in the subspace spanned by the [math] eigenvectors. The partitioning is encoded in a matrix [math] in indicator form, which is computed by approximating the eigenvector matrix by a product of [math] and an orthogonal matrix. A measure of the distance of a graph to being [math]-partitionable is defined, as well as two cut (cost) functions, for which Cheeger inequalities are proved; thus the relation between the eigenvalue and partitioning problems is established. Numerical examples are given that demonstrate that the partitioning algorithm is efficient and robust.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"7 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 134-147, March 2024. Abstract. The elastic potential is a valuable modeling tool for many applications, including medical imaging. One reason for this is that the energy and its Gâteaux derivative, the elastic operator, have strong coupling properties. Although these properties are desirable from a modeling perspective, they are not advantageous from a computational or operator decomposition perspective. In this paper, we show that the elastic operator can be spectrally decomposed despite its coupling property when equipped with sliding boundary conditions. Moreover, we present a discretization that is fully compatible with this spectral decomposition. In particular, for image registration problems, this decomposition opens new possibilities for multispectral solution techniques and fine-tuned operator-based regularization.
{"title":"The Spectral Decomposition of the Continuous and Discrete Linear Elasticity Operators with Sliding Boundary Conditions","authors":"Jan Modersitzki","doi":"10.1137/22m1541320","DOIUrl":"https://doi.org/10.1137/22m1541320","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 134-147, March 2024. <br/> Abstract. The elastic potential is a valuable modeling tool for many applications, including medical imaging. One reason for this is that the energy and its Gâteaux derivative, the elastic operator, have strong coupling properties. Although these properties are desirable from a modeling perspective, they are not advantageous from a computational or operator decomposition perspective. In this paper, we show that the elastic operator can be spectrally decomposed despite its coupling property when equipped with sliding boundary conditions. Moreover, we present a discretization that is fully compatible with this spectral decomposition. In particular, for image registration problems, this decomposition opens new possibilities for multispectral solution techniques and fine-tuned operator-based regularization.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"82 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 84-111, March 2024. Abstract. This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial-symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples demonstrate theoretical properties and computational efficiency of the SCF and its acceleration.
{"title":"Variational Characterization of Monotone Nonlinear Eigenvector Problems and Geometry of Self-Consistent Field Iteration","authors":"Zhaojun Bai, Ding Lu","doi":"10.1137/22m1525326","DOIUrl":"https://doi.org/10.1137/22m1525326","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 84-111, March 2024. <br/> Abstract. This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial-symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples demonstrate theoretical properties and computational efficiency of the SCF and its acceleration.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"36 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 59-83, March 2024. Abstract. Structure-preserving doubling algorithms (SDAs) are efficient algorithms for solving Riccati-type matrix equations. However, breakdowns may occur in SDAs. To remedy this drawback, in this paper, we first introduce [math]-symplectic forms ([math]-SFs), consisting of symplectic matrix pairs with a Hermitian parametric matrix [math]. Based on [math]-SFs, we develop modified SDAs (MSDAs) for solving the associated Riccati-type equations. MSDAs generate sequences of symplectic matrix pairs in [math]-SFs and prevent breakdowns by employing a reasonably selected Hermitian matrix [math]. In practical implementations, we show that the Hermitian matrix [math] in MSDAs can be chosen as a real diagonal matrix that can reduce the computational complexity. The numerical results demonstrate a significant improvement in the accuracy of the solutions by MSDAs.
{"title":"Structure-Preserving Doubling Algorithms That Avoid Breakdowns for Algebraic Riccati-Type Matrix Equations","authors":"Tsung-Ming Huang, Yueh-Cheng Kuo, Wen-Wei Lin, Shih-Feng Shieh","doi":"10.1137/23m1551791","DOIUrl":"https://doi.org/10.1137/23m1551791","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 59-83, March 2024. <br/> Abstract. Structure-preserving doubling algorithms (SDAs) are efficient algorithms for solving Riccati-type matrix equations. However, breakdowns may occur in SDAs. To remedy this drawback, in this paper, we first introduce [math]-symplectic forms ([math]-SFs), consisting of symplectic matrix pairs with a Hermitian parametric matrix [math]. Based on [math]-SFs, we develop modified SDAs (MSDAs) for solving the associated Riccati-type equations. MSDAs generate sequences of symplectic matrix pairs in [math]-SFs and prevent breakdowns by employing a reasonably selected Hermitian matrix [math]. In practical implementations, we show that the Hermitian matrix [math] in MSDAs can be chosen as a real diagonal matrix that can reduce the computational complexity. The numerical results demonstrate a significant improvement in the accuracy of the solutions by MSDAs.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"51 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139415016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 24-58, March 2024. Abstract. The cross-product matrix-based CJ-FEAST SVDsolver proposed previously by the authors is shown to compute the left singular vector possibly much less accurately than the right singular vector and may be numerically backward unstable when a desired singular value is small. In this paper, an alternative augmented matrix-based CJ-FEAST SVDsolver is proposed to compute the singular triplets of a large matrix [math] with the singular values in an interval [math] contained in the singular spectrum. The new CJ-FEAST SVDsolver is a subspace iteration applied to an approximate spectral projector of the augmented matrix [math] associated with the eigenvalues in [math], and it constructs approximate left and right singular subspaces independently, onto which [math] is projected to obtain the Ritz approximations to the desired singular triplets. Compact estimates are given for the accuracy of the approximate spectral projector constructed by the Chebyshev–Jackson series expansion in terms of series degree, and a number of convergence results are established. The new solver is proved to be always numerically backward stable. A convergence comparison of the cross-product-based and augmented matrix-based CJ-FEAST SVDsolvers is made, and a general-purpose choice strategy between the two solvers is proposed for the robustness and overall efficiency. Numerical experiments confirm all the results and meanwhile demonstrate that the proposed solver is more robust and substantially more efficient than the corresponding contour integral-based versions that exploit the trapezoidal rule and the Gauss–Legendre quadrature to construct an approximate spectral projector.
{"title":"An Augmented Matrix-Based CJ-FEAST SVDsolver for Computing a Partial Singular Value Decomposition with the Singular Values in a Given Interval","authors":"Zhongxiao Jia, Kailiang Zhang","doi":"10.1137/23m1547500","DOIUrl":"https://doi.org/10.1137/23m1547500","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 24-58, March 2024. <br/> Abstract. The cross-product matrix-based CJ-FEAST SVDsolver proposed previously by the authors is shown to compute the left singular vector possibly much less accurately than the right singular vector and may be numerically backward unstable when a desired singular value is small. In this paper, an alternative augmented matrix-based CJ-FEAST SVDsolver is proposed to compute the singular triplets of a large matrix [math] with the singular values in an interval [math] contained in the singular spectrum. The new CJ-FEAST SVDsolver is a subspace iteration applied to an approximate spectral projector of the augmented matrix [math] associated with the eigenvalues in [math], and it constructs approximate left and right singular subspaces independently, onto which [math] is projected to obtain the Ritz approximations to the desired singular triplets. Compact estimates are given for the accuracy of the approximate spectral projector constructed by the Chebyshev–Jackson series expansion in terms of series degree, and a number of convergence results are established. The new solver is proved to be always numerically backward stable. A convergence comparison of the cross-product-based and augmented matrix-based CJ-FEAST SVDsolvers is made, and a general-purpose choice strategy between the two solvers is proposed for the robustness and overall efficiency. Numerical experiments confirm all the results and meanwhile demonstrate that the proposed solver is more robust and substantially more efficient than the corresponding contour integral-based versions that exploit the trapezoidal rule and the Gauss–Legendre quadrature to construct an approximate spectral projector.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"30 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 1-23, March 2024. Abstract. The implicit trace estimation problem asks for an approximation of the trace of a square matrix, accessed via matrix-vector products (matvecs). This paper designs new randomized algorithms, XTrace and XNysTrace, for the trace estimation problem by exploiting both variance reduction and the exchangeability principle. For a fixed budget of matvecs, numerical experiments show that the new methods can achieve errors that are orders of magnitude smaller than existing algorithms, such as the Girard–Hutchinson estimator or the Hutch++ estimator. A theoretical analysis confirms the benefits by offering a precise description of the performance of these algorithms as a function of the spectrum of the input matrix. The paper also develops an exchangeable estimator, XDiag, for approximating the diagonal of a square matrix using matvecs.
{"title":"XTrace: Making the Most of Every Sample in Stochastic Trace Estimation","authors":"Ethan N. Epperly, Joel A. Tropp, Robert J. Webber","doi":"10.1137/23m1548323","DOIUrl":"https://doi.org/10.1137/23m1548323","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 1-23, March 2024. <br/> Abstract. The implicit trace estimation problem asks for an approximation of the trace of a square matrix, accessed via matrix-vector products (matvecs). This paper designs new randomized algorithms, XTrace and XNysTrace, for the trace estimation problem by exploiting both variance reduction and the exchangeability principle. For a fixed budget of matvecs, numerical experiments show that the new methods can achieve errors that are orders of magnitude smaller than existing algorithms, such as the Girard–Hutchinson estimator or the Hutch++ estimator. A theoretical analysis confirms the benefits by offering a precise description of the performance of these algorithms as a function of the spectrum of the input matrix. The paper also develops an exchangeable estimator, XDiag, for approximating the diagonal of a square matrix using matvecs.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"30 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ming Zhou, Merico Argentati, Andrew V. Knyazev, Klaus Neymeyr
SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1852-1878, December 2023. Abstract. Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and closely related research on properties of matrix-based signal filters are challenging and are attracting increased attention due to their recent applications in spectral data clustering and graph-based signal processing. We combine majorization-based techniques pioneered for investigating the Rayleigh–Ritz method in [A. V. Knyazev and M. E. Argentati, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521–1537] with tools of classical analysis of the block power method by Rutishauser [Numer. Math., 13 (1969), pp. 4–13] to derive sharp convergence rate bounds of abstract block iterations, wherein tuples of tangents of principal angles or relative errors of Ritz values are bounded using majorization in terms of arranged partial sums and tuples of convergence factors. Our novel bounds are robust in the presence of clusters of eigenvalues, improve previous results, and are applicable to most known block iterative solvers and matrix-based filters, e.g., to block power, Chebyshev, and Lanczos methods combined with polynomial filtering. The sharpness of our bounds is fundamental, implying that the bounds cannot be improved without further assumptions.
{"title":"Sharp Majorization-Type Cluster Robust Bounds for Block Filters and Eigensolvers","authors":"Ming Zhou, Merico Argentati, Andrew V. Knyazev, Klaus Neymeyr","doi":"10.1137/23m1551729","DOIUrl":"https://doi.org/10.1137/23m1551729","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1852-1878, December 2023. <br/> Abstract. Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and closely related research on properties of matrix-based signal filters are challenging and are attracting increased attention due to their recent applications in spectral data clustering and graph-based signal processing. We combine majorization-based techniques pioneered for investigating the Rayleigh–Ritz method in [A. V. Knyazev and M. E. Argentati, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521–1537] with tools of classical analysis of the block power method by Rutishauser [Numer. Math., 13 (1969), pp. 4–13] to derive sharp convergence rate bounds of abstract block iterations, wherein tuples of tangents of principal angles or relative errors of Ritz values are bounded using majorization in terms of arranged partial sums and tuples of convergence factors. Our novel bounds are robust in the presence of clusters of eigenvalues, improve previous results, and are applicable to most known block iterative solvers and matrix-based filters, e.g., to block power, Chebyshev, and Lanczos methods combined with polynomial filtering. The sharpness of our bounds is fundamental, implying that the bounds cannot be improved without further assumptions.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"228 7","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1822-1851, December 2023. Abstract. We provide a characterization of the continuous-time Markov models where the Markov matrices from the model can be parameterized directly in terms of the associated rate matrices (generators). That is, each Markov matrix can be expressed as the sum of the identity matrix and a rate matrix from the model. We show that the existence of an underlying Jordan algebra provides a sufficient condition, which becomes necessary for (so-called) linear models. We connect this property to the well-known uniformization procedure for continuous-time Markov chains by demonstrating that the property is equivalent to all Markov matrices from the model taking the same form as the corresponding discrete-time Markov matrices in the uniformized process. We apply our results to analyze two model hierarchies practically important to phylogenetic inference, obtained by assuming (i) time reversibility and (ii) permutation symmetry, respectively.
{"title":"Uniformization Stable Markov Models and Their Jordan Algebraic Structure","authors":"Luke Cooper, Jeremy Sumner","doi":"10.1137/22m1474527","DOIUrl":"https://doi.org/10.1137/22m1474527","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1822-1851, December 2023. <br/> Abstract. We provide a characterization of the continuous-time Markov models where the Markov matrices from the model can be parameterized directly in terms of the associated rate matrices (generators). That is, each Markov matrix can be expressed as the sum of the identity matrix and a rate matrix from the model. We show that the existence of an underlying Jordan algebra provides a sufficient condition, which becomes necessary for (so-called) linear models. We connect this property to the well-known uniformization procedure for continuous-time Markov chains by demonstrating that the property is equivalent to all Markov matrices from the model taking the same form as the corresponding discrete-time Markov matrices in the uniformized process. We apply our results to analyze two model hierarchies practically important to phylogenetic inference, obtained by assuming (i) time reversibility and (ii) permutation symmetry, respectively.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"2 2","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1879-1907, December 2023. Abstract. We solve the open problem of describing the possible Kronecker invariants of quasi-regular matrix pencils under bounded rank perturbations. By a quasi-regular matrix pencil we mean the full (normal) rank matrix pencil. The solution is explicit and constructive, and it is valid over arbitrary fields.
{"title":"Bounded Rank Perturbations of Quasi-Regular Pencils Over Arbitrary Fields","authors":"Marija Dodig, Marko Stošić","doi":"10.1137/22m1504068","DOIUrl":"https://doi.org/10.1137/22m1504068","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1879-1907, December 2023. <br/> Abstract. We solve the open problem of describing the possible Kronecker invariants of quasi-regular matrix pencils under bounded rank perturbations. By a quasi-regular matrix pencil we mean the full (normal) rank matrix pencil. The solution is explicit and constructive, and it is valid over arbitrary fields.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Philipp Dettling, Roser Homs, Carlos Améndola, Mathias Drton, Niels Richard Hansen
SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1799-1821, December 2023. Abstract. The recently introduced graphical continuous Lyapunov models provide a new approach to statistical modeling of correlated multivariate data. The models view each observation as a one-time cross-sectional snapshot of a multivariate dynamic process in equilibrium. The covariance matrix for the data is obtained by solving a continuous Lyapunov equation that is parametrized by the drift matrix of the dynamic process. In this context, different statistical models postulate different sparsity patterns in the drift matrix, and it becomes a crucial problem to clarify whether a given sparsity assumption allows one to uniquely recover the drift matrix parameters from the covariance matrix of the data. We study this identifiability problem by representing sparsity patterns by directed graphs. Our main result proves that the drift matrix is globally identifiable if and only if the graph for the sparsity pattern is simple (i.e., does not contain directed 2-cycles). Moreover, we present a necessary condition for generic identifiability and provide a computational classification of small graphs with up to 5 nodes.
{"title":"Identifiability in Continuous Lyapunov Models","authors":"Philipp Dettling, Roser Homs, Carlos Améndola, Mathias Drton, Niels Richard Hansen","doi":"10.1137/22m1520311","DOIUrl":"https://doi.org/10.1137/22m1520311","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1799-1821, December 2023. <br/> Abstract. The recently introduced graphical continuous Lyapunov models provide a new approach to statistical modeling of correlated multivariate data. The models view each observation as a one-time cross-sectional snapshot of a multivariate dynamic process in equilibrium. The covariance matrix for the data is obtained by solving a continuous Lyapunov equation that is parametrized by the drift matrix of the dynamic process. In this context, different statistical models postulate different sparsity patterns in the drift matrix, and it becomes a crucial problem to clarify whether a given sparsity assumption allows one to uniquely recover the drift matrix parameters from the covariance matrix of the data. We study this identifiability problem by representing sparsity patterns by directed graphs. Our main result proves that the drift matrix is globally identifiable if and only if the graph for the sparsity pattern is simple (i.e., does not contain directed 2-cycles). Moreover, we present a necessary condition for generic identifiability and provide a computational classification of small graphs with up to 5 nodes.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"230 12","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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