Generalized low-rank approximation to the symmetric positive semidefinite matrix

Haixia Chang; Chunmei Li; Longsheng Liu; Haixia Chang; Chunmei Li; Longsheng Liu

doi:10.3934/math.2025368

AIMS Mathematics

2025, Volume 10, Issue 4: 8022-8035. doi: 10.3934/math.2025368

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Generalized low-rank approximation to the symmetric positive semidefinite matrix

1.
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
2.
College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China
3.
School of Mathematics and Physics, Anqing Normal University, Anqing 246011, China

Received: 27 November 2024 Revised: 10 March 2025 Accepted: 25 March 2025 Published: 07 April 2025
MSC : 15A33, 65F30, 65K10, 68W25

In this paper, we consider the generalized low-rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $ \underset{X}{\min} \sum^{m}_{i = 1}\left \Vert A_{i}-B_{i}XB_{i}^{T}\right \Vert^{2}_{F}, $ where $ X $ is an unknown symmetric positive semidefinite matrix whose rank is less than or equal to a positive integer $ k $. We first characterize the feasible set as $ X = YY^{T} $, where $ Y $ has the order $ n\times k $, and then convert the generalized low-rank approximation into an unconstrained generalized optimization problem. Finally, we employ the nonlinear conjugate gradient method with an exact line search to solve the generalized optimization problem. We also give numerical examples to exemplify the results.
- generalized low-rank approximation,
- symmetric positive semidefinite matrix,
- generalized optimization,
- nonlinear conjugate gradient method,
- feasible set
Citation: Haixia Chang, Chunmei Li, Longsheng Liu. Generalized low-rank approximation to the symmetric positive semidefinite matrix[J]. AIMS Mathematics, 2025, 10(4): 8022-8035. doi: 10.3934/math.2025368

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Abstract

In this paper, we consider the generalized low-rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $ \underset{X}{\min} \sum^{m}_{i = 1}\left \Vert A_{i}-B_{i}XB_{i}^{T}\right \Vert^{2}_{F}, $ where $ X $ is an unknown symmetric positive semidefinite matrix whose rank is less than or equal to a positive integer $ k $. We first characterize the feasible set as $ X = YY^{T} $, where $ Y $ has the order $ n\times k $, and then convert the generalized low-rank approximation into an unconstrained generalized optimization problem. Finally, we employ the nonlinear conjugate gradient method with an exact line search to solve the generalized optimization problem. We also give numerical examples to exemplify the results.

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