Tensor Completion using Kronecker Rank-1 Tensor Train with Application to Visual Data Inpainting

The problem of data reconstruction with partly sampled elements under a tensor structure, which is referred to as tensor completion, is addressed in this work. The properties of the rank-1 tensor train decomposition and the tensor Kronecker decomposition are introduced at first, and then the tensor Kronecker rank as well as Kronecker rank-1 tensor train decomposition are defined. The general tensor completion idea is presented following the criterion of minimizing the number of Kronecker rank- 1 tensors, which is relaxed to the thresholding problem and the solution is derived. Furthermore, the number of Kronecker rank-1 tensors that the proposed algorithm can retrieve and its complexity order are analyzed. Computer simulations are carried out on real visual data sets and demonstrate that our method yields a superior performance over the state-of-the-art approaches in terms of recovery accuracy and/or computational complexity.