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A New Matrix Maximization Model for Computing Ratios of Generalized Singular Values from High-Order GSVD

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Abstract

A High-Order Generalized Singular Value Decomposition (HO-GSVD) is employed to compare multiple matrices \(\{ \textbf{A}_i \}_{i=1}^{N}\) with different row dimensions by using their generalized singular values \(\{\sigma _{i,k} \}_{i=1}^{N}\) respectively. The ratio values of \(\sigma _{i,k} / \sigma _{j,k}\) can be used to indicate the significance of the k-th basis vector of the right hand side of matrix from HO-GSVD for multiple matrices \(\{\textbf{A}_i \}_{i=1}^{N}\). The main aim of this paper is to propose and study a new matrix maximization model for computing ratios of \(\sigma _{i,k} / \sigma _{j,k}\) from \({\textbf{A}}_{1},\ldots ,{\textbf{A}}_{N}\). The resulting optimization problem can be solved by using Newton method on Lie Groups, and the convergence of the Newton method with well defined initial value can also be established. Numerical examples for synthetic data and mRNA expression data sets are reported to demonstrate the fast performance of the proposed method for solving the optimization model with other existing state-of-the-art algorithms and Riemannian Newton method.

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  1. https://doi.org/10.1371/journal.pone.0028072

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Wei-Wei Xu

  2. Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

    Michael K. Ng

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  1. Wei-Wei Xu
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  2. Michael K. Ng
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The research of Wei-Wei Xu is partially supported by the National Natural Science Foundation of China (No. 11971243). The research of Michael K. Ng is partially supported by Hong Kong Research Grant Council GRF 12300218, 12300519, 17201020, 17300021, C1013-21GF, C7004-21GF and Joint NSFC-RGC N-HKU76921.

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Xu, WW., Ng, M.K. A New Matrix Maximization Model for Computing Ratios of Generalized Singular Values from High-Order GSVD. J Sci Comput 94, 35 (2023). https://doi.org/10.1007/s10915-022-02071-8

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