Robust to Rank Selection: Low-Rank Sparse Tensor-Ring Completion.

Tensor-ring (TR) decomposition was recently studied and applied for low-rank tensor completion due to its powerful representation ability of high-order tensors. However, most of the existing TR-based methods tend to suffer from deterioration when the selected rank is larger than the true one. To address this issue, this article proposes a new low-rank sparse TR completion method by imposing the Frobenius norm regularization on its latent space. Specifically, we theoretically establish that the proposed method is capable of exploiting the low rankness and Kronecker-basis-representation (KBR)-based sparsity of the target tensor using the Frobenius norm of latent TR-cores. We optimize the proposed TR completion by block coordinate descent (BCD) algorithm and design a modified TR decomposition for the initialization of this algorithm. Extensive experimental results on synthetic data and visual data have demonstrated that the proposed method is able to achieve better results compared to the conventional TR-based completion methods and other state-of-the-art methods and, meanwhile, is quite robust even if the selected TR-rank increases.