A separable surrogate function method for sparse and low-rank matrices decomposition

Abstract

Suppose we have an observed data matrix D which can be decomposed as the sum of a sparse matrix E and a low-rank matrix A. The purpose of this paper is to recover the sparse component and low-rank component individually from a given observation matrix. In this paper, a novel method proposed here deviates from the other approaches listed is the splitting of the constraint $D=A+E$ into two constraints, $A=A_0$ and $E=E_0$, where $A_0$ and $E_0$ are the real low-rank and sparse part of data matrix D. This separation strategy is referred to as the separable surrogate function (SSF). In such case, two iterative methods are designed to solve this problem. Correspondingly, the convergence analysis of these two iterative methods is given respectively. Simulations about real-data examples and applications on images decomposition show the feasibility and effectiveness of the proposed algorithms.

Keywords:

• Sparse matrix
• low-rank matrix
• robust principal component analysis
• image decomposition

2010 Mathematics Subject Classifications:

• 65F30
• 65K05
• 90C25

Acknowledgements

The authors wish to thank the anonymous referees and the editors for providing their valuable comments which have significantly improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the National Statistical Science Research Project of China under grant No. 2018LZ23, the Key Research Projects of Henan Higher Education Institutions of China under grant Nos. 20A110014 and 20A120004, and the National Natural Science Foundation of China under grant No. 11671318.