Auto-weighted robust low-rank tensor completion via tensor-train

Abstract

Nowadays, multi-dimensional data (tensor data) have shown their capability of preserving multilinear structures. Due to the measuring error or other non-human factors, these data often suffer signal corruptions or missing values, or even both. To address these issues simultaneously, this paper studies the Robust Tensor Completion (RTC) problem, a mixed problem of the known Low-Rank Tensor Completion (LRTC) and Robust Principal Component Analysis (RPCA). Based on Tensor-Train rank (TT rank), the proposed model is able to capture the latent structure information of tensor data by recovering the low-rank component and separating the sparse component from the partial observations. To make TT rank more effective, an auto-weighted mechanism is utilized to balance the importance of different matricizations from the same tensor. We also propose a more flexible tensor augmentation approach called Tree Ket Augmentation (Tree-KA) to obtain a higher-order tensor from a lower one with it a new general explanation. Alternating direction method of multipliers (ADMM) is employed to solve the resulting model and extensive numerical experiments have verified the effectiveness of the proposed model compared with other state-of-the-art methods.