Low‐rank tensor completion for visual data recovery via the tensor train rank‐1 decomposition

Abstract

In this study, the authors study the problem of tensor completion, in particular for three-dimensional arrays such as visual data. Previous works have shown that the low-rank constraint can produce impressive performances for tensor completion. These works are often solved by means of Tucker rank. However, Tucker rank does not capture the intrinsic correlation of the tensor entries. Therefore, the authors propose a new proximal operator for the approximation of tensor nuclear norms based on tensor-train rank-1 decomposition via the singular value decomposition. The proximal operator will perform a soft-thresholding operation on tensor singular values. In addition, the low-rank constraint can capture the global structure of data well, but it does not exploit local smooth of visual data. Therefore, they integrate total variation as a regularisation term into low-rank tensor completion. Finally, they use a primal–dual splitting to achieve optimisation. Experimental results have shown that the proposed method, can preserve the multi-dimensional nature inherent in the data, and thus provide superior results over many state-of-the-art tensor completion techniques.