关键词: 自适应变换, 非凸松弛, 近端交替最小化, 张量补全

Abstract: The common point of many tensor completion methods is to firstly project the tensor into the transformed domain by a pre-defined transform, and then describe the low-rankness or sparsity of the tensor in the transformed domain (shortened to the transformed tensor). However, the pre-defined transform is not general. To address this problem, a new tensor average rank under self-adaptive transforms is firstly proposed, where the transformed tensor is eventually solved by continuous iterations, and the newly defined tensor average rank is an extension of the tensor average rank under invertible linear transforms. Then, this paper proposes a tensor completion model based on self-adaptive transforms and non-convex relaxation. The self-adaptation means that the transformed tensor is the unknown tensor to be solved, and it can continuously adjust itself based on the observed tensor during the process of minimizing the objective function until it becomes the optimal solution of the objective function. The model uses a non-convex surrogate to approximate the tensor average rank under self-adaptive transforms and adopts the  norm to measure the sparsity of the transformed tensor. In the process of solving the optimal solution through the proximal alternating minimization framework, the model adaptively learns the transformed low-rank tensor and the transformed sparse tensor based on the observed tensor, and then converts the transformed low-rank tensor and the transformed sparse tensor into the original space through the learned transform matrices, respectively. Finally, the completed tensor is obtained. Experiments are carried out on grey-scale videos, multispectral images and hyperspectral images. The experimental results demonstrate that the proposed method further improves the completion performance compared with other representative tensor completion methods.

Key words: self-adaptive transforms, non-convex relaxation, proximal alternating minimization, tensor completion