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.