Low-Rank Tensor Completion by Sum of Tensor Nuclear Norm Minimization

In this paper, we study the problem of low-rank tensor completion with the purpose of recovering a low-rank tensor from a tensor with partial observed items. To date, there are several different definitions of tensor ranks. We focus the study on the low tubal rank tensor completion task. Previous works solve the low tubal rank tensor completion/recovery problems by convex tensor nuclear norm minimization. However, this kind of tensor nuclear norm is orientation dependent, which is originally due to the definition of tensor-tensor product. Based on the convex tensor nuclear norm minimization, the tensor recovery performance varies when the orientation of the input data is different. However, in practice, it is generally hard to choose the best way of the data input. To address this issue, we propose a new convex model which is based on the sum of tensor nuclear norm minimization. It includes the existing tensor nuclear norm minimization model as a special case which is corresponding to an orientation of the input data. The proposed model is convex and thus can be solved efficiently. Numerical experiments on images and video sequences demonstrate the effectiveness of our proposed method.