Tensor ring decomposition-based model with interpretable gradient factors regularization for tensor completion

Tensor ring (TR) decomposition, which factorizes a tensor into a sequence of cyclically interconnected third-order TR factors, is a powerful tool to capture the global low-rankness of high-dimensional data. However, the understanding of the physical interpretation of TR factors is not clear. In this paper, we first empirically discover the physical interpretation of TR factors in the gradient domain (termed as gradient factors) and then give the theoretical justification. Based on the interpretable gradient factors, we suggest a TR decomposition-based model with interpretable gradient factors regularization (TR-GFR) for tensor completion. To be specific, we consider the low-rankness and transformed sparsity priors of gradient factors to boost the performance and robustness of TR decomposition-based model. In addition, we develop an effective proximal alternating minimization algorithm to solve the proposed model. Numerical experiments validate that the proposed TR-GFR is superior to the compared state-of-the-art methods in terms of PSNR and SSIM values and more robust with TR rank.