A non-convex tensor rank approximation for tensor completion

Low-rankness has been widely exploited for the tensor completion problem. Recent advances have suggested that the tensor nuclear norm often leads to a promising approximation for the tensor rank. It treats the singular values equally to pursue the convexity of the objective function, while the singular values for the practical images have clear physical meanings with different importance and should be treated differently. In this paper, we propose a non-convex logDet function as a smooth approximation for tensor rank instead of the convex tensor nuclear norm and introduce it into the low-rank tensor completion problem. An alternating direction method of multiplier (ADMM)-based method is developed to solve the problem. Experimental results have shown that the proposed method can significantly outperform existing state-of-the-art nuclear norm-based methods for tensor completion.