Nonconvex tensor rank minimization and its applications to tensor recovery

Low-rank tensor recovery (LRTR) has recently emerged as the potent tools for representing multidimensional data. One of the most popular LRTR is tensor rank minimization that acts as the estimating tensor rank for a given tensor. However, the existing convex tensor rank approximation methods suffer from the serious rank estimation bias due to neglecting the physical meanings of singular values along each mode. In this paper, we propose a new method to approximate the tensor rank by using the nonconvex logarithmic surrogate function of the singular values, and the redefined rank approximation can further reduce to a convex weighted nuclear norm minimization (WNNM) problem. By embedding the tensor rank function into the tensor completion (TC) and tensor robust PCA (TRPCA) frameworks, new models are formulated to enhance tensor processing. Additionally, by introducing relaxation forms of the proposed tensor rank function, the alternating direction method of multipliers (ADMM) can be adopted for the models. The proposed nonconvex tensor rank minimization method can achieve state-of-the-art performance in tensor recovery, including tensor completion and background subtraction.