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

Wei Wei Xu; Michael K. Ng

doi:10.1007/s10915-022-02071-8

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

Wei Wei Xu^*, Michael K. Ng

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

Abstract

A High-Order Generalized Singular Value Decomposition (HO-GSVD) is employed to compare multiple matrices {Ai}^N_i=1 with different row dimensions by using their generalized singular values {σi,k}^N_i=1 respectively. The ratio values of σ_i_,_k/ σ_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 {Ai}^N_i=1. The main aim of this paper is to propose and study a new matrix maximization model for computing ratios of σ_i_,_k/ σ_j_,_k from A₁, … , 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.

Original language	English
Article number	35
Number of pages	22
Journal	Journal of Scientific Computing
Volume	94
Issue number	2
Early online date	6 Jan 2023
DOIs	https://doi.org/10.1007/s10915-022-02071-8
Publication status	Published - Feb 2023

Scopus Subject Areas

Theoretical Computer Science
Software
Numerical Analysis
General Engineering
Computational Mathematics
Computational Theory and Mathematics
Applied Mathematics

User-Defined Keywords

Global and quadratic convergence
High-order generalized singular value decomposition
Matrix maximization models
Newton methods on Lie groups

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10.1007/s10915-022-02071-8

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

abstract = "A High-Order Generalized Singular Value Decomposition (HO-GSVD) is employed to compare multiple matrices {Ai}Ni=1 with different row dimensions by using their generalized singular values {σi,k}Ni=1 respectively. The ratio values of σi,k/ σj,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 {Ai}Ni=1. The main aim of this paper is to propose and study a new matrix maximization model for computing ratios of σi,k/ σj,k from A1, … , AN. 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.",

keywords = "Global and quadratic convergence, High-order generalized singular value decomposition, Matrix maximization models, Newton methods on Lie groups",

author = "Xu, {Wei Wei} and Ng, {Michael K.}",

note = "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. Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

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doi = "10.1007/s10915-022-02071-8",

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N2 - A High-Order Generalized Singular Value Decomposition (HO-GSVD) is employed to compare multiple matrices {Ai}Ni=1 with different row dimensions by using their generalized singular values {σi,k}Ni=1 respectively. The ratio values of σi,k/ σj,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 {Ai}Ni=1. The main aim of this paper is to propose and study a new matrix maximization model for computing ratios of σi,k/ σj,k from A1, … , AN. 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.

AB - A High-Order Generalized Singular Value Decomposition (HO-GSVD) is employed to compare multiple matrices {Ai}Ni=1 with different row dimensions by using their generalized singular values {σi,k}Ni=1 respectively. The ratio values of σi,k/ σj,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 {Ai}Ni=1. The main aim of this paper is to propose and study a new matrix maximization model for computing ratios of σi,k/ σj,k from A1, … , AN. 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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A New Matrix Maximization Model for Computing Ratios of Generalized Singular Values from High-Order GSVD

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