A singular value shrinkage thresholding algorithm for folded concave penalized low-rank matrix optimization problems

In this paper, we study the low-rank matrix optimization problem, where the loss function is smooth but not necessarily convex, and the penalty term is a nonconvex (folded concave) continuous relaxation of the rank function. Firstly, we give the closed-form singular value shrinkage thresholding operators for several matrix-valued folded concave penalty functions. Secondly, we adopt a singular value shrinkage thresholding (SVST) algorithm for the nonconvex low-rank matrix optimization problem, and prove that the proposed SVST algorithm converges to a stationary point of the problem. Furthermore, we show that the limit point satisfies a global necessary optimality condition which can exclude too many stationary points even local minimizers in order to refine the solutions. We conduct a large number of numeric experiments to test the performance of SVST algorithm on the randomly generated low-rank matrix completion problem, the real 2D and 3D image recovery problem and the multivariate linear regression problem. Numerical results show that SVST algorithm is very competitive for low-rank matrix optimization problems in comparison with some state-of-the-art algorithms.