Low-Rank Tensor Completion by Approximating the Tensor Average Rank

Abstract

To alleviate the bias generated by the ℓ1-norm in the low-rank tensor completion problem, nonconvex surrogates/regularizers have been suggested to replace the tensor nuclear norm, although both can achieve sparsity. However, the thresholding functions of these nonconvex regularizers may not have closed-form expressions and thus iterations are needed, which increases the computational loads. To solve this issue, we devise a framework to generate sparsity-inducing regularizers with closed-form thresholding functions. These regularizers are applied to low-tubal-rank tensor completion, and efficient algorithms based on the alternating direction method of multipliers are developed. Furthermore, convergence of our methods is analyzed and it is proved that the generated sequences are bounded and any limit point is a stationary point. Experimental results using synthetic and real-world datasets show that the proposed algorithms outperform the state-of-the-art methods in terms of restoration performance.

A novel $ \ell_{0} $ minimization framework of tensor tubal rank and its multi-dimensional image completion application

“…Thus, the tensor with low tubal rank always has low average rank. Besides, the minimization of tensor tubal rank can be approximated by minimizing the tensor average rank in applications [38]. Hence, we formulate an equivalent form of minimizing tensor average rank to approximate the tensor tubal rank.…”