Multi-stage convex relaxation method for low-rank and sparse matrix separation problem

The low-rank and sparse matrix separation problem, recovering a low-rank matrix and a sparse matrix from a small number of linear observations, has arisen in a broad set of applications. The desire for efficiency motivated the research on it, however, most of them focus on the nuclear norm plus ℓ1-norm minimization problem, which is a single convex relaxation of the low-rank and sparse matrix separation problem. This paper is devoted to investigate a multi-stage convex relaxation method to recover the low-rank and sparse components via solving a sequence of convex relaxations, while the first stage of the method includes the nuclear norm plus ℓ1-norm minimization problem. Numerical studies indicate that the multi-stage convex relaxation outperforms the one-stage convex relaxation, especially when observations are incomplete and noisy. We have also verified the efficiency of the proposed method by applying it to the static background extraction from surveillance video.