A Nonconvex Relaxation Approach to Low-Rank Tensor Completion

Low-rank tensor completion plays an important role in many applications such as image processing, computer vision, and machine learning. A widely used convex relaxation of this problem is to minimize the nuclear norm of the square deal matrix generated by reshaping a tensor. However, this approach can be substantially suboptimal. In order to seek a highly accurate solution, in this paper, we propose to use a family of nonconvex functions onto the singular values of the square deal matrix of the tensor to approximate the rank of the tensor. A proximal linearized minimization (PLM) algorithm is proposed to solve the resulting model. Furthermore, based on the Kurdyka-Łojasiewicz property, we show that the sequence generated by the PLM algorithm globally converges to a critical point of the objective function. Extensive numerical experiments including synthetic data, video data, and the extended Yale Face Database B show the effectiveness of the proposed model compared with several existing state-of-the-art models.