A local convex method for rank-sparsity factorization

•A novel local convex method is proposed for low-rank-sparsity matrix factorization.•It uses a local convex envelope to approximate the cardinality function of matrix.•Two models are discussed with implicit or explicit rank restrictions, respectively.•ADMM methods are discussed for solving the local problems.•Iterative improvement and refinement methods are discussed for improving solutions.

A novel method is proposed for recovering low-rank component and sparsity component of noisy observations, using a local convex envelope of the matrix cardinality function over a local box. Two local relaxation models combined with implicit or explicit rank restriction are proposed for solving the rank-sparsity factorization. An iterative approach of the local relaxation and a post-processing refinement are also given to further improve the factorization, together with updating rules of the local box. Numerical examples show the efficiency of the proposed methods in two applications.