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

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 = A₀ and E = E₀, where A₀ and E₀ 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.