An evolutionary multiobjective method for low-rank and sparse matrix decomposition

Tao Wu,Maoguo Gong,Jiao Shi,Yu Lei

doi:10.3934/bdia.2017006

Tao Wu, Maoguo Gong + Show 2 more

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This paper addresses the problem of finding the low-rank and sparse components of a given matrix. The problem involves two conflicting objective functions, reducing the rank and sparsity of each part simultaneously. Previous methods combine two objectives into a single objective penalty function to solve with traditional numerical optimization approaches. The main contribution of this paper is to put forward a multiobjective method to decompose the given matrix into low-rank component and sparse part. We optimize two objective functions with an evolutionary multiobjective algorithm MOEA/D. Another contribution of this paper, a modified low-rank and sparse matrix model is proposed, which simplifying the variable of objective functions and improving the efficiency of multiobjective optimization. The proposed method obtains a set of solutions with different trade-off between low-rank and sparse objectives, and decision makers can choose one or more satisfied decomposed results according to different requirements directly. Experiments conducted on artificial datasets and nature images, show that the proposed method always obtains satisfied results, and the convergence, stability and robustness of the proposed method is acceptable.

Highlights

Matrix decomposition can catch some characteristic information of matrix, in which low- rank and sparse components are two of the most interesting
We model low-rank and sparse matrix decomposition as a multiobjective problem (MOP), and modify the multiobjective LRSMD (MOLRSMD), which has simple structure and is easy to optimize
We introduce the model of multiobjective low-rank and sparse matrix decomposition based on matrix singular value decomposition

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Introduction

Matrix decomposition can catch some characteristic information of matrix, in which low- rank and sparse components are two of the most interesting. We model low-rank and sparse matrix decomposition as a multiobjective problem (MOP), and modify the multiobjective LRSMD (MOLRSMD), which has simple structure and is easy to optimize. We will describe the motivation of multiobjective low-rank and sparse matrix decomposition.

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