SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 847-874, June 2024. Abstract. Hierarchical matrices are dense but data-sparse matrices that use low-rank factorizations of suitable submatrices to reduce the storage and computational cost to linear-polylogarithmic complexity. In this paper, we propose a new approach to efficiently compute QR factorizations in the hierarchical matrix format based on block Householder transformations. To prevent unnecessarily high ranks in the resulting factors and to increase speed and accuracy, the algorithm meticulously tracks for which intermediate results low-rank factorizations are available. We also use a special storage scheme for the block Householder reflector to further reduce computational and storage costs. Numerical tests for two- and three-dimensional Laplacian boundary element matrices, different radial basis function kernel matrices, and matrices of typical hierarchical matrix structures but filled with random entries illustrate the performance of the new algorithm in comparison to some other QR algorithms for hierarchical matrices from the literature.
SIAM 矩阵分析与应用期刊》，第 45 卷，第 2 期，第 847-874 页，2024 年 6 月。 摘要层次矩阵是高密度但数据稀疏的矩阵，它使用合适子矩阵的低秩因子来将存储和计算成本降低到线性-多对数复杂度。在本文中，我们提出了一种基于分块豪斯赫德（Householder）变换的新方法，以高效计算分层矩阵格式中的 QR 因式分解。为了防止计算出的因数出现不必要的高阶，并提高速度和准确性，该算法会仔细跟踪哪些中间结果可以进行低阶因式分解。我们还为块豪斯赫德反射器采用了一种特殊的存储方案，以进一步降低计算和存储成本。对二维和三维拉普拉斯边界元素矩阵、不同径向基函数核矩阵以及典型分层矩阵结构但充满随机条目的矩阵进行的数值测试，说明了新算法与文献中其他一些针对分层矩阵的 QR 算法相比的性能。
{"title":"A Block Householder–Based Algorithm for the QR Decomposition of Hierarchical Matrices","authors":"Vincent Griem, Sabine Le Borne","doi":"10.1137/22m1544555","DOIUrl":"https://doi.org/10.1137/22m1544555","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 847-874, June 2024. <br/> Abstract. Hierarchical matrices are dense but data-sparse matrices that use low-rank factorizations of suitable submatrices to reduce the storage and computational cost to linear-polylogarithmic complexity. In this paper, we propose a new approach to efficiently compute QR factorizations in the hierarchical matrix format based on block Householder transformations. To prevent unnecessarily high ranks in the resulting factors and to increase speed and accuracy, the algorithm meticulously tracks for which intermediate results low-rank factorizations are available. We also use a special storage scheme for the block Householder reflector to further reduce computational and storage costs. Numerical tests for two- and three-dimensional Laplacian boundary element matrices, different radial basis function kernel matrices, and matrices of typical hierarchical matrix structures but filled with random entries illustrate the performance of the new algorithm in comparison to some other QR algorithms for hierarchical matrices from the literature.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"101 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140627286","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 2, Page 829-846, June 2024. Abstract. Computer implementations of vector orthogonalization algorithms produce a sequence of supposedly orthogonal vectors, but rounding-errors can cause loss of orthogonality and rank. Nevertheless these computational algorithms can be very effective as parts of various methods. We develop a general theory based on the augmented orthogonal matrix developed in [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 565–583] that can be applied to any such algorithm. This can be combined with a rounding-error analysis of the algorithm to analyze its finite-precision behavior. We apply this combination to prove that a particular Lanczos tridiagonalization of a Hermitian matrix always computes components for which backward-stable solutions to [math], [math], exist. If an appropriate rounding-error analysis is available, the approach can apparently be applied to any computation producing a sequence of supposedly orthogonal [math]-vectors, where a linear combination of these vectors is intended to approximate some quantity.
{"title":"Analyzing Vector Orthogonalization Algorithms","authors":"Christopher C. Paige","doi":"10.1137/22m1519523","DOIUrl":"https://doi.org/10.1137/22m1519523","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 829-846, June 2024. <br/> Abstract. Computer implementations of vector orthogonalization algorithms produce a sequence of supposedly orthogonal vectors, but rounding-errors can cause loss of orthogonality and rank. Nevertheless these computational algorithms can be very effective as parts of various methods. We develop a general theory based on the augmented orthogonal matrix developed in [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 565–583] that can be applied to any such algorithm. This can be combined with a rounding-error analysis of the algorithm to analyze its finite-precision behavior. We apply this combination to prove that a particular Lanczos tridiagonalization of a Hermitian matrix always computes components for which backward-stable solutions to [math], [math], exist. If an appropriate rounding-error analysis is available, the approach can apparently be applied to any computation producing a sequence of supposedly orthogonal [math]-vectors, where a linear combination of these vectors is intended to approximate some quantity.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"51 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140581516","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 801-827, March 2024. Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [math] in a larger unitary transformation [math] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [math], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.
{"title":"Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices","authors":"Daan Camps, Lin Lin, Roel Van Beeumen, Chao Yang","doi":"10.1137/22m1484298","DOIUrl":"https://doi.org/10.1137/22m1484298","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 801-827, March 2024. <br/> Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [math] in a larger unitary transformation [math] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [math], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"18 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140106580","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}
Matthias Christandl, Fulvio Gesmundo, Vladimir Lysikov, Vincent Steffan
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 771-800, March 2024. Abstract. Tensors are often studied by introducing preorders such as restriction and degeneration. The former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local linear maps may vary along a curve, and the resulting tensor is expressed as a limit along this curve. In this work, we introduce and study partial degeneration, a special version of degeneration where one of the local linear maps is constant while the others vary along a curve. Motivated by algebraic complexity, quantum entanglement, and tensor networks, we present constructions based on matrix multiplication tensors and find examples by making a connection to the theory of prehomogeneous tensor spaces. We highlight the subtleties of this new notion by showing obstruction and classification results for the unit tensor. To this end, we study the notion of aided rank, a natural generalization of tensor rank. The existence of partial degenerations gives strong upper bounds on the aided rank of a tensor, which allows one to turn degenerations into restrictions. In particular, we present several examples, based on the W-tensor and the Coppersmith–Winograd tensors, where lower bounds on aided rank provide obstructions to the existence of certain partial degenerations.
{"title":"Partial Degeneration of Tensors","authors":"Matthias Christandl, Fulvio Gesmundo, Vladimir Lysikov, Vincent Steffan","doi":"10.1137/23m1554898","DOIUrl":"https://doi.org/10.1137/23m1554898","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 771-800, March 2024. <br/> Abstract. Tensors are often studied by introducing preorders such as restriction and degeneration. The former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local linear maps may vary along a curve, and the resulting tensor is expressed as a limit along this curve. In this work, we introduce and study partial degeneration, a special version of degeneration where one of the local linear maps is constant while the others vary along a curve. Motivated by algebraic complexity, quantum entanglement, and tensor networks, we present constructions based on matrix multiplication tensors and find examples by making a connection to the theory of prehomogeneous tensor spaces. We highlight the subtleties of this new notion by showing obstruction and classification results for the unit tensor. To this end, we study the notion of aided rank, a natural generalization of tensor rank. The existence of partial degenerations gives strong upper bounds on the aided rank of a tensor, which allows one to turn degenerations into restrictions. In particular, we present several examples, based on the W-tensor and the Coppersmith–Winograd tensors, where lower bounds on aided rank provide obstructions to the existence of certain partial degenerations.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"87 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140097723","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 744-770, March 2024. Abstract. We consider the solution of large stiff systems of ODEs with explicit exponential Runge–Kutta integrators. These problems arise from semidiscretized semilinear parabolic PDEs on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of [math]-functions in exponential integrators to the approximation of the action of a smaller number of matrix exponentials on certain vectors. State-of-the-art computational methods use polynomial Krylov subspaces of adaptive size for this task. They have the drawback that the required number of Krylov subspace iterations to obtain a desired tolerance increases drastically with the spectral radius of the discrete linear differential operator, e.g., the problem size. We present an approach that leverages rational Krylov subspace methods promising superior approximation qualities. We prove a novel a posteriori error estimate of rational Krylov approximations to the action of the matrix exponential on vectors for single time points, which allows for an adaptive approach similar to existing polynomial Krylov techniques. We discuss pole selection and the efficient solution of the arising sequences of shifted linear systems by direct and preconditioned iterative solvers. Numerical experiments show that our method outperforms the state of the art for sufficiently large spectral radii of the discrete linear differential operators. The key to this are approximately constant numbers of rational Krylov iterations, which enable a near-linear scaling of the runtime with respect to the problem size.
{"title":"Adaptive Rational Krylov Methods for Exponential Runge–Kutta Integrators","authors":"Kai Bergermann, Martin Stoll","doi":"10.1137/23m1559439","DOIUrl":"https://doi.org/10.1137/23m1559439","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 744-770, March 2024. <br/> Abstract. We consider the solution of large stiff systems of ODEs with explicit exponential Runge–Kutta integrators. These problems arise from semidiscretized semilinear parabolic PDEs on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of [math]-functions in exponential integrators to the approximation of the action of a smaller number of matrix exponentials on certain vectors. State-of-the-art computational methods use polynomial Krylov subspaces of adaptive size for this task. They have the drawback that the required number of Krylov subspace iterations to obtain a desired tolerance increases drastically with the spectral radius of the discrete linear differential operator, e.g., the problem size. We present an approach that leverages rational Krylov subspace methods promising superior approximation qualities. We prove a novel a posteriori error estimate of rational Krylov approximations to the action of the matrix exponential on vectors for single time points, which allows for an adaptive approach similar to existing polynomial Krylov techniques. We discuss pole selection and the efficient solution of the arising sequences of shifted linear systems by direct and preconditioned iterative solvers. Numerical experiments show that our method outperforms the state of the art for sufficiently large spectral radii of the discrete linear differential operators. The key to this are approximately constant numbers of rational Krylov iterations, which enable a near-linear scaling of the runtime with respect to the problem size.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"32 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140035180","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}
Huan He, Ziyuan Tang, Shifan Zhao, Yousef Saad, Yuanzhe Xi
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 712-743, March 2024. Abstract. This paper develops a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as with inexact Newton methods—depending on which variant is implemented. We prove theoretically and verify experimentally, on a variety of problems from simulation experiments to deep learning applications, that our method is a powerful accelerated iterative algorithm. The code is available at https://github.com/Data-driven-numerical-methods/Nonlinear-Truncated-Conjugate-Residual.
{"title":"nlTGCR: A Class of Nonlinear Acceleration Procedures Based on Conjugate Residuals","authors":"Huan He, Ziyuan Tang, Shifan Zhao, Yousef Saad, Yuanzhe Xi","doi":"10.1137/23m1576360","DOIUrl":"https://doi.org/10.1137/23m1576360","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 712-743, March 2024. <br/> Abstract. This paper develops a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as with inexact Newton methods—depending on which variant is implemented. We prove theoretically and verify experimentally, on a variety of problems from simulation experiments to deep learning applications, that our method is a powerful accelerated iterative algorithm. The code is available at https://github.com/Data-driven-numerical-methods/Nonlinear-Truncated-Conjugate-Residual.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"51 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006558","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}
Zachary M. Boyd, Nicolas Fraiman, Jeremy L. Marzuola, Peter J. Mucha, Braxton Osting
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 685-711, March 2024. Abstract. We provide a rearrangement based algorithm for detection of subgraphs of k vertices with long escape times for directed or undirected networks that is not combinatorially complex to compute. Complementing other notions of densest subgraphs and graph cuts, our method is based on the mean hitting time required for a random walker to leave a designated set and hit the complement. We provide a new relaxation of this notion of hitting time on a given subgraph and use that relaxation to construct a subgraph detection algorithm that can be computed easily and a generalization to K-partitioning schemes. Using a modification of the subgraph detector on each component, we propose a graph partitioner that identifies regions where random walks live for comparably large times. Importantly, our method implicitly respects the directed nature of the data for directed graphs while also being applicable to undirected graphs. We apply the partitioning method for community detection to a large class of models and real-world data sets.
SIAM 矩阵分析与应用期刊》，第 45 卷，第 1 期，第 685-711 页，2024 年 3 月。 摘要。我们提供了一种基于重排的算法，用于检测有向或无向网络中逃逸时间较长的 k 个顶点的子图，其计算并不复杂。作为对其他最密子图和图切割概念的补充，我们的方法基于随机漫步者离开指定集合并命中补集所需的平均命中时间。我们对给定子图上的命中时间这一概念进行了新的松弛，并利用这一松弛构建了一种可以轻松计算的子图检测算法，并将其推广到 K 分区方案中。利用对每个组件上的子图检测器的修改，我们提出了一种图分割器，它能识别随机游走存活时间相当大的区域。重要的是，我们的方法隐含地尊重了有向图数据的有向性，同时也适用于无向图。我们将群落检测的分区方法应用于一大类模型和真实世界的数据集。
{"title":"An Escape Time Formulation for Subgraph Detection and Partitioning of Directed Graphs","authors":"Zachary M. Boyd, Nicolas Fraiman, Jeremy L. Marzuola, Peter J. Mucha, Braxton Osting","doi":"10.1137/23m1553790","DOIUrl":"https://doi.org/10.1137/23m1553790","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 685-711, March 2024. <br/> Abstract. We provide a rearrangement based algorithm for detection of subgraphs of k vertices with long escape times for directed or undirected networks that is not combinatorially complex to compute. Complementing other notions of densest subgraphs and graph cuts, our method is based on the mean hitting time required for a random walker to leave a designated set and hit the complement. We provide a new relaxation of this notion of hitting time on a given subgraph and use that relaxation to construct a subgraph detection algorithm that can be computed easily and a generalization to K-partitioning schemes. Using a modification of the subgraph detector on each component, we propose a graph partitioner that identifies regions where random walks live for comparably large times. Importantly, our method implicitly respects the directed nature of the data for directed graphs while also being applicable to undirected graphs. We apply the partitioning method for community detection to a large class of models and real-world data sets.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"134 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139968531","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 661-684, March 2024. Abstract. Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing this task. RJD applies a standard eigenvalue solver to random linear combinations of the matrices. Unlike existing optimization-based methods, RJD is simple to implement and leverages existing high-quality linear algebra software packages. Our main novel contribution is to prove robust recovery: Given a family that is [math]-near to a commuting family, RJD jointly diagonalizes this family, with high probability, up to an error of norm [math]. We also discuss how the algorithm can be further improved by deflation techniques and demonstrate its state-of-the-art performance by numerical experiments with synthetic and real-world data.
{"title":"Randomized Joint Diagonalization of Symmetric Matrices","authors":"Haoze He, Daniel Kressner","doi":"10.1137/22m1541265","DOIUrl":"https://doi.org/10.1137/22m1541265","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 661-684, March 2024. <br/> Abstract. Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing this task. RJD applies a standard eigenvalue solver to random linear combinations of the matrices. Unlike existing optimization-based methods, RJD is simple to implement and leverages existing high-quality linear algebra software packages. Our main novel contribution is to prove robust recovery: Given a family that is [math]-near to a commuting family, RJD jointly diagonalizes this family, with high probability, up to an error of norm [math]. We also discuss how the algorithm can be further improved by deflation techniques and demonstrate its state-of-the-art performance by numerical experiments with synthetic and real-world data.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"133 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969714","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 634-660, March 2024. Abstract. Matrix factorizations in dual number algebra, a hypercomplex number system, have been applied to kinematics, spatial mechanisms, and other fields recently. We develop an approach to identify spatiotemporal patterns in the brain such as traveling waves using the singular value decomposition (SVD) of dual matrices in this paper. Theoretically, we propose the compact dual singular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well as a necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we report on the optimal solution to the best rank-[math] approximation under a newly defined quasi-metric in the dual complex number system. The CDSVD is also related to the dual Moore–Penrose generalized inverse. Numerically, comparisons with other available algorithms are conducted, which indicate less computational costs of our proposed CDSVD. In addition, the infinitesimal part of the CDSVD can identify the true rank of the original matrix from the noise-added matrix, but the classical SVD cannot. Next, we employ experiments on simulated time-series data and a road monitoring video to demonstrate the beneficial effect of the infinitesimal parts of dual matrices in spatiotemporal pattern identification. Finally, we apply this approach to the large-scale brain functional magnetic resonance imaging data, identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function.
{"title":"Singular Value Decomposition of Dual Matrices and its Application to Traveling Wave Identification in the Brain","authors":"Tong Wei, Weiyang Ding, Yimin Wei","doi":"10.1137/23m1556642","DOIUrl":"https://doi.org/10.1137/23m1556642","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 634-660, March 2024. <br/> Abstract. Matrix factorizations in dual number algebra, a hypercomplex number system, have been applied to kinematics, spatial mechanisms, and other fields recently. We develop an approach to identify spatiotemporal patterns in the brain such as traveling waves using the singular value decomposition (SVD) of dual matrices in this paper. Theoretically, we propose the compact dual singular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well as a necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we report on the optimal solution to the best rank-[math] approximation under a newly defined quasi-metric in the dual complex number system. The CDSVD is also related to the dual Moore–Penrose generalized inverse. Numerically, comparisons with other available algorithms are conducted, which indicate less computational costs of our proposed CDSVD. In addition, the infinitesimal part of the CDSVD can identify the true rank of the original matrix from the noise-added matrix, but the classical SVD cannot. Next, we employ experiments on simulated time-series data and a road monitoring video to demonstrate the beneficial effect of the infinitesimal parts of dual matrices in spatiotemporal pattern identification. Finally, we apply this approach to the large-scale brain functional magnetic resonance imaging data, identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"23 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756211","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}
Alice Cortinovis, Daniel Kressner, Yuji Nakatsukasa
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 619-633, March 2024. Abstract. This work is concerned with the computation of the action of a matrix function f(A), such as the matrix exponential or the matrix square root, on a vector b. For a general matrix A, this can be done by computing the compression of A onto a suitable Krylov subspace. Such compression is usually computed by forming an orthonormal basis of the Krylov subspace using the Arnoldi method. In this work, we propose to compute (nonorthonormal) bases in a faster way and to use a fast randomized algorithm for least-squares problems to compute the compression of A onto the Krylov subspace. We present some numerical examples which show that our algorithms can be faster than the standard Arnoldi method while achieving comparable accuracy.
SIAM 矩阵分析与应用期刊》，第 45 卷，第 1 期，第 619-633 页，2024 年 3 月。 摘要。这项工作涉及矩阵函数 f(A) 对向量 b 的作用的计算，例如矩阵指数或矩阵平方根。对于一般矩阵 A，可以通过计算 A 对合适的 Krylov 子空间的压缩来实现。这种压缩通常是通过使用 Arnoldi 方法形成 Krylov 子空间的正交基来计算的。在这项工作中，我们建议以更快的方式计算（非正态）基，并使用最小二乘问题的快速随机算法来计算 A 到 Krylov 子空间的压缩。我们给出了一些数值示例，表明我们的算法比标准阿诺德方法更快，同时精度相当。
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