A smoothing proximal gradient algorithm for matrix rank minimization problem

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimizers and the same optimal value...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 81; no. 2; pp. 519 - 538
Main Authors: Yu, Quan, Zhang, Xinzhen
Format: Journal Article
Language:English
Published: New York Springer US 01.03.2022
Springer Nature B.V
Subjects:
Algorithms
Convex and Discrete Geometry
Lower bounds
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Smoothing
Statistics
15A83
90C30
65K05
Low-rank approximation
Nonsmooth convex loss function
Smoothing method
15A03
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimizers and the same optimal value. In particular, we introduce a class of lifted stationary points of the relaxed problem and show that any local minimizer of the relaxed problem must be a lifted stationary point. In addition, we derive lower bound property for the nonzero singular values of the lifted stationary point and hence also of the local minimizers of the relaxed problem. Then the smoothing proximal gradient (SPG) algorithm is proposed to find a lifted stationary point of the continuous relaxation model. Moreover, it is shown that any accumulating point of the sequence generated by SPG algorithm is a lifted stationary point. At last, numerical examples show the efficiency of the SPG algorithm.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-021-00337-9