A Nonconvex Method to Low-Rank Matrix Completion

In recent years, the problem of recovering a low-rank matrix from partial entries, known as low-rank matrix completion problem, has attracted much attention in many applications. However, it is a NP-hard problem due to the nonconvexity nature of the matrix rank function. In this paper, a rank Laplace function is studied to recover the low-rank matrices. Firstly, we propose an iterative Laplace thresholding algorithm to solve the regularized Laplace low-rank matrix completion problem. Secondly, some other iterative thresholding algorithms are designed to recover the low-rank matrices. Finally, we provide a series of numerical simulations to test the proposed algorithms on some low-rank matrix completion and image inpainting problems, and the results show that our algorithms perform better than some state-of-art methods in recovering the low-rank matrices.