Tensor Wheel Decomposition for Robust Low-Rank Tensor Completion

Abstract

The robust tensor completion (RTC) problem, which aims to recover a low-rank tensor from partially observed data contaminated by noise, has attracted growing attention. Recently, methods based on various tensor decompositions, such as tensor train (TT) decomposition, tensor ring (TR) decomposition, and fully-connected tensor network (FCTN) decomposition, have made notable progress in RTC. However, TT and TR neglect potential interactions between nonadjacent factors, and the number of hyperparameters in the FCTN model scales quadratically with the tensor order, making it challenging to specify an optimal set of FCTN ranks for higher-order tensors. To address these issues, we introduce the recently proposed tensor wheel (TW) decomposition and propose a TW decomposition–based optimization model (RTC-TW) for the RTC problem. The proposed model preserves the interactions between any two factors while ensuring that the number of TW ranks grows only linearly with tensor order, thereby effectively mitigating the aforementioned limitations. We develop an efficient algorithm to solve the model based on proximal alternating minimization (PAM) with a theoretical convergence guarantee. Comprehensive numerical experiments demonstrate that our model achieves state-of-the-art performance compared to existing low-rank methods.

Keywords: Robust tensor completion, Tensor wheel (TW) decomposition, Proximal alternating minimization (PAM)