Low-rank matrix recovery via novel double nonconvex nonsmooth rank minimization with ADMM

In recent years, using the low-rank structure of the data to recover the matrix has become one of the hot issues in image processing. However, it usually leads to a suboptimal solution because current convex relaxations of the rank function (e.g., nuclear norm, etc.) reduce the rank components excessively and treat each rank component equally. To solve this problem, many nonconvex relaxations have been proposed, but their convergence properties are generally not easy to guarantee. In this paper, we construct a novel double nonconvex nonsmooth model by combining the truncated Schatten- p norm and the Schatten- p norm for the first time, called N-DNNR. It ignores the effect of large singular values on the matrix rank through the truncated Schatten- p norm and approximates the rank function by the Schatten- p norm. The alternating directional multiplier method (ADMM) is introduced to solve the N-DNNR model, which is more robust than other state-of-the-art algorithms and guarantees global convergence. In particular, updating the variable X is a nonconvex optimization problem, and we solve it by the approximate gradient algorithm (PG) and the weighted singular value threshold operator (WSVF), which has a closed-form solution. Finally, experimental results on synthetic data and real images show that the N-DNNR model has stronger generalization and recovery capabilities than the TSPN model.