A fast non-convex regularizer for low rank matrix completion
Several non-convex surrogates for rank functions were proposed recently in the rank minimization for the corrupted image recovery and data analysis. In this paper, we propose a fast non-convex regularizer whose main feature is piecewise linearity with three thresholds. Since most singular values are low or infinitesimal for low-rank matrices, which is beneficial for threshold comparisons, the proximal gradient descent for the proposed regularizer can be fast. We further propose that the randomized singular value decomposition (SVD) algorithm can be adopted to perform the partial SVD. When the matrix is low-rank, the proximal gradient descent with the randomized SVD can effectively recover the low-rank structure and save computational time. Experiments on matrix recovery and image recovery are conducted to validate that the proposed algorithm outperforms state-of-the-art regularizers and other low-rank recovery methods.