Tensor Recovery With Weighted Tensor Average Rank

In this article, a curious phenomenon in the tensor recovery algorithm is considered: can the same recovered results be obtained when the observation tensors in the algorithm are transposed in different ways? If not, it is reasonable to imagine that some information within the data will be lost for the case of observation tensors under certain transpose operators. To solve this problem, a new tensor rank called weighted tensor average rank (WTAR) is proposed to learn the relationship between different resulting tensors by performing a series of transpose operators on an observation tensor. WTAR is applied to three-order tensor robust principal component analysis (TRPCA) to investigate its effectiveness. Meanwhile, to balance the effectiveness and solvability of the resulting model, a generalized model that involves the convex surrogate and a series of nonconvex surrogates are studied, and the corresponding worst case error bounds of the recovered tensor is given. Besides, a generalized tensor singular value thresholding (GTSVT) method and a generalized optimization algorithm based on GTSVT are proposed to solve the generalized model effectively. The experimental results indicate that the proposed method is effective.