{"title":"An Apocalypse-Free First-Order Low-Rank Optimization Algorithm with at Most One Rank Reduction Attempt per Iteration","authors":"Guillaume Olikier, P.-A. Absil","doi":"10.1137/22m1518256","DOIUrl":"https://doi.org/10.1137/22m1518256","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060490","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}
Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M=P_1SP_2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We discuss some mathematics properties of CoS-NMF, and present the relationships with other related matrix factorizations such as CUR decomposition, generalized separable NMF(GS-NMF), and bi-orthogonal tri-factorization (BiOR-NM3F). An optimization model for CoS-NMF is proposed and alternated fast gradient method is employed to solve the model. Numerical experiments on synthetic datasets, document datasets and facial databases are conducted to verify the effectiveness of our CoS-NMF model. Compared to state-of-the-art methods, CoS-NMF model performs very well in co-clustering task, and preserves a good approximation to the input data matrix as well.
{"title":"Coseparable Nonnegative Matrix Factorization","authors":"Junjun Pan, Michael K. Ng","doi":"10.1137/22m1510509","DOIUrl":"https://doi.org/10.1137/22m1510509","url":null,"abstract":"Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M=P_1SP_2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We discuss some mathematics properties of CoS-NMF, and present the relationships with other related matrix factorizations such as CUR decomposition, generalized separable NMF(GS-NMF), and bi-orthogonal tri-factorization (BiOR-NM3F). An optimization model for CoS-NMF is proposed and alternated fast gradient method is employed to solve the model. Numerical experiments on synthetic datasets, document datasets and facial databases are conducted to verify the effectiveness of our CoS-NMF model. Compared to state-of-the-art methods, CoS-NMF model performs very well in co-clustering task, and preserves a good approximation to the input data matrix as well.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353866","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}
Vivak Patel, Mohammad Jahangoshahi, Daniel Adrian Maldonado
. Randomized linear solvers leverage randomization to structure-blindly compress and solve a linear system to produce an inexpensive solution. While such a property is highly desirable, randomized linear solvers often suffer when it comes to performance as either (1) problem structure is not being exploited, and (2) hardware is inefficiently used. Thus, randomized adaptive solvers are starting to appear that use the benefits of randomness while attempting to still exploit problem structure and reduce hardware inefficiencies. Unfortunately, such randomized adaptive solvers are likely to be without a theoretical foundation to show that they will work (i.e., find a solution). Accordingly, here, we distill three general criteria for randomized block adaptive solvers, which, as we show, will guarantee convergence of the randomized adaptive solver and supply a worst-case rate of convergence. We will demonstrate that these results apply to existing randomized block adaptive solvers, and to several that we devise for demonstrative purposes.
{"title":"Randomized Block Adaptive Linear System Solvers","authors":"Vivak Patel, Mohammad Jahangoshahi, Daniel Adrian Maldonado","doi":"10.1137/22m1488715","DOIUrl":"https://doi.org/10.1137/22m1488715","url":null,"abstract":". Randomized linear solvers leverage randomization to structure-blindly compress and solve a linear system to produce an inexpensive solution. While such a property is highly desirable, randomized linear solvers often suffer when it comes to performance as either (1) problem structure is not being exploited, and (2) hardware is inefficiently used. Thus, randomized adaptive solvers are starting to appear that use the benefits of randomness while attempting to still exploit problem structure and reduce hardware inefficiencies. Unfortunately, such randomized adaptive solvers are likely to be without a theoretical foundation to show that they will work (i.e., find a solution). Accordingly, here, we distill three general criteria for randomized block adaptive solvers, which, as we show, will guarantee convergence of the randomized adaptive solver and supply a worst-case rate of convergence. We will demonstrate that these results apply to existing randomized block adaptive solvers, and to several that we devise for demonstrative purposes.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64315847","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}
When analyzing complex networks, an important task is the identification of those nodes which play a leading role for the overall communicability of the network. In the context of modifying networks (or making them robust against targeted attacks or outages), it is also relevant to know how sensitive the network’s communicability reacts to changes in certain nodes or edges. Recently, the concept of total network sensitivity was introduced in [O. De la Cruz Cabrera, J. Jin, S. Noschese, and L. Reichel, Appl. Numer. Math., 172 (2022) pp. 186–205], which allows one to measure how sensitive the total communicability of a network is to the addition or removal of certain edges. One shortcoming of this concept is that sensitivities are extremely costly to compute when using a straightforward approach (orders of magnitude more expensive than the corresponding communicability measures). In this work, we present computational procedures for estimating network sensitivity with a cost that is essentially linear in the number of nodes for many real-world complex networks. Additionally, we extend the sensitivity concept such that it also covers sensitivity of subgraph centrality and the Estrada index, and we discuss the case of node removal. We propose a priori bounds for these sensitivities which capture well the qualitative behavior and give insight into the general behavior of matrix function based network indices under perturbations. These bounds are based on decay results for Fréchet derivatives of matrix functions with structured, low-rank direction terms which might be of independent interest also for applications other than network analysis.
在分析复杂网络时，一个重要的任务是识别对网络整体通信起主导作用的节点。在修改网络(或使其对目标攻击或中断具有健壮性)的上下文中，了解网络的可通信性对某些节点或边缘的变化的反应敏感程度也是相关的。最近，在[0]中引入了全网络灵敏度的概念。De la Cruz Cabrera, J. Jin, S. Noschese和L. Reichel, apple。号码。数学。， 172 (2022) pp. 186-205]，它允许人们测量网络的总通信能力对某些边的添加或移除有多敏感。这个概念的一个缺点是，当使用直接的方法时，灵敏度的计算成本非常高(比相应的通信度量要高几个数量级)。在这项工作中，我们提出了用于估计网络灵敏度的计算程序，其成本在许多现实世界的复杂网络的节点数量中基本上是线性的。此外，我们扩展了灵敏度概念，使其涵盖了子图中心性和Estrada指数的灵敏度，并讨论了节点移除的情况。我们提出了这些敏感性的先验界，它很好地捕捉了定性行为，并深入了解了基于矩阵函数的网络指标在扰动下的一般行为。这些边界是基于矩阵函数的fr导数的衰减结果，这些函数具有结构化的、低秩的方向项，这对于除网络分析以外的应用也可能是独立的。
{"title":"Sensitivity of Matrix Function Based Network Communicability Measures: Computational Methods and A Priori Bounds","authors":"Marcel Schweitzer","doi":"10.1137/23m1556708","DOIUrl":"https://doi.org/10.1137/23m1556708","url":null,"abstract":"When analyzing complex networks, an important task is the identification of those nodes which play a leading role for the overall communicability of the network. In the context of modifying networks (or making them robust against targeted attacks or outages), it is also relevant to know how sensitive the network’s communicability reacts to changes in certain nodes or edges. Recently, the concept of total network sensitivity was introduced in [O. De la Cruz Cabrera, J. Jin, S. Noschese, and L. Reichel, Appl. Numer. Math., 172 (2022) pp. 186–205], which allows one to measure how sensitive the total communicability of a network is to the addition or removal of certain edges. One shortcoming of this concept is that sensitivities are extremely costly to compute when using a straightforward approach (orders of magnitude more expensive than the corresponding communicability measures). In this work, we present computational procedures for estimating network sensitivity with a cost that is essentially linear in the number of nodes for many real-world complex networks. Additionally, we extend the sensitivity concept such that it also covers sensitivity of subgraph centrality and the Estrada index, and we discuss the case of node removal. We propose a priori bounds for these sensitivities which capture well the qualitative behavior and give insight into the general behavior of matrix function based network indices under perturbations. These bounds are based on decay results for Fréchet derivatives of matrix functions with structured, low-rank direction terms which might be of independent interest also for applications other than network analysis.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135890679","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}
{"title":"Analyzing the Influence of Agents in Trust Networks: Applying Nonsmooth Eigensensitivity Theory to a Graph Centrality Problem","authors":"Jon Donnelly, Peter G. Stechlinski","doi":"10.1137/21m146884x","DOIUrl":"https://doi.org/10.1137/21m146884x","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64315182","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}
Vanni Noferini, Lauri Nyman, Javier Pérez, María C. Quintana
Zeros of rational transfer function matrices are the eigenvalues of associated polynomial system matrices under minimality conditions. In this paper, we define a structured condition number for a simple eigenvalue of a (locally) minimal polynomial system matrix , which in turn is a simple zero of its transfer function matrix . Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yields a structured perturbation theory for simple zeros of rational matrices . To capture all the zeros of , regardless of whether they are poles, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of being not a pole of since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur’s unstructured condition number for eigenvalues of matrix polynomials and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.
{"title":"Perturbation Theory of Transfer Function Matrices","authors":"Vanni Noferini, Lauri Nyman, Javier Pérez, María C. Quintana","doi":"10.1137/22m1509825","DOIUrl":"https://doi.org/10.1137/22m1509825","url":null,"abstract":"Zeros of rational transfer function matrices are the eigenvalues of associated polynomial system matrices under minimality conditions. In this paper, we define a structured condition number for a simple eigenvalue of a (locally) minimal polynomial system matrix , which in turn is a simple zero of its transfer function matrix . Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yields a structured perturbation theory for simple zeros of rational matrices . To capture all the zeros of , regardless of whether they are poles, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of being not a pole of since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur’s unstructured condition number for eigenvalues of matrix polynomials and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136081554","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}
{"title":"Eigenvalue Embedding of Damped Vibroacoustic System with No-Spillover","authors":"K. Zhao, Zhong Y. Liu","doi":"10.1137/22m1527416","DOIUrl":"https://doi.org/10.1137/22m1527416","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76183506","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}
In this paper, we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices , , with coefficients in a generic field and with common column dimension . We give necessary and sufficient conditions for a matrix to be a GCRD using the Smith normal form of the compound matrix obtained by concatenating vertically, where . We also describe the complete set of degrees of freedom for the solution , and we link it to the Smith form and Hermite form of . We then give an algorithm for constructing a particular minimum size solution for this problem when or , using state-space techniques. This new method works directly on the coefficient matrices of , using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of .
{"title":"Revisiting the Matrix Polynomial Greatest Common Divisor","authors":"Vanni Noferini, Paul Van Dooren","doi":"10.1137/22m1531993","DOIUrl":"https://doi.org/10.1137/22m1531993","url":null,"abstract":"In this paper, we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices , , with coefficients in a generic field and with common column dimension . We give necessary and sufficient conditions for a matrix to be a GCRD using the Smith normal form of the compound matrix obtained by concatenating vertically, where . We also describe the complete set of degrees of freedom for the solution , and we link it to the Smith form and Hermite form of . We then give an algorithm for constructing a particular minimum size solution for this problem when or , using state-space techniques. This new method works directly on the coefficient matrices of , using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of .","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135795066","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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