Two-stage convex relaxation approach to least squares loss constrained low-rank plus sparsity optimization problems

This paper is concerned with the least squares loss constrained low-rank plus sparsity optimization problems that seek a low-rank matrix and a sparse matrix by minimizing a positive combination of the rank function and the zero norm over a least squares constraint set describing the observation or prior information on the target matrix pair. For this class of NP-hard optimization problems, we propose a two-stage convex relaxation approach by the majorization for suitable locally Lipschitz continuous surrogates, which have a remarkable advantage in reducing the error yielded by the popular nuclear norm plus $$\ell _1$$l1-norm convex relaxation method. Also, under a suitable restricted eigenvalue condition, we establish a Frobenius norm error bound for the optimal solution of each stage and show that the error bound of the first stage convex relaxation (i.e. the nuclear norm plus $$\ell _1$$l1-norm convex relaxation), can be reduced much by the second stage convex relaxation, thereby providing the theoretical guarantee for the two-stage convex relaxation approach. We also verify the efficiency of the proposed approach by applying it to some random test problems and some problems with real data arising from specularity removal from face images, and foreground/background separation from surveillance videos.