Alternating Direction Method of Multipliers for Sparse and Low-Rank Decomposition Based on Nonconvex Nonsmooth Weighted Nuclear Norm

Sparse and low-rank decomposition (SLRD) poses a big challenge in many fields. The existing methods are used to solve SLRD problem via formulating approximations of sparse and low-rank matrices. These conventional methods consider the approximation of the low-rank matrix as its nuclear norm, which is a convex surrogate function of the rank. Since these approaches simultaneously minimize all the singular values, and thus the rank may not be well approximated in practice. In this paper, we extend the nonconvex nonsmooth weighted nuclear norm to approximate the low-rank matrix and formulate a general form nonconvex nonsmooth sparse and low-rank matrices decomposition problem. Hence, we can adopt the alternating direction method of multipliers to solve this nonconvex nonsmooth problem and analyze its convergence. Simulation results and discussions are given to validate the proposed method.