Low-rank and sparse matrix decomposition via the truncated nuclear norm and a sparse regularizer

Abstract

Recovering the low-rank and sparse components from a given matrix is a challenging problem that has many real applications. This paper proposes a novel algorithm to address this problem by introducing a sparse prior on the low-rank component. Specifically, the low-rank component is assumed to be sparse in a transform domain and a sparse regularizer formulated as an $$\ell _1$$ -norm term is employed to promote the sparsity. The truncated nuclear norm is used to model the low-rank prior, rather than the nuclear norm used in most existing methods, to achieve a better approximation to the rank of the considered matrix. Furthermore, an efficient solving method based on a two-stage iterative scheme is developed to address the raised optimization problem. The proposed algorithm is applied to deal with synthetic data and real applications including face image shadow removal and video background subtraction, and the experimental results validate the effectiveness and accuracy of the proposed approach as compared with other methods.