A new surrogate for tensor multirank and applications in image and video completion

How to define and relax the tensor rank is a challenging and meaningful topic. The CP-rank, n-rank, and tensor multirank are three of the most popular definitions. Among them n-rank and tensor multirank are widely studied in the low-rank tensor completion problem, and their relaxations are sum of nuclear norm (SNN) and tensor nuclear norm (TNN), respectively. Both the two kinds of nuclear norm treat the singular values equally, while the different singular values for the practical images represent different physical meanings and should be treated differently. In this paper, we propose a tensor logDet function as the relaxation for tensor multirank rather than TNN. To demonstrate the effectiveness of the proposed function, we introduce the function into the low-rank tensor completion problem and develop an alternating direction method of multipliers (ADMM)-based method. Extensive experiments have shown that the proposed method outperforms the SNN and TNN based methods.