A Generalized Non-convex Method for Robust Tensor Completion

In this paper, we concentrate on the robust tensor completion (RTC) problem, which aims to recover a low-rank tensor from partial observations corrupted by sparse noise. Most existing methods for RTC utilize the tensor nuclear norm (TNN) to evaluate the tensor rank. However, the TNN often yields biased solutions due to its loose approximation for the tensor rank. To solve this problem, we derive a unified non-convex surrogate that better approximates the tensor rank. Our surrogate is composed of several non-convex penalty functions. Further, we propose a generalized non-convex model, which minimizes a weighted combination of the unified non-convex surrogate and the t1-norm data fidelity term. To solve the proposed model, we devise a simple but efficient algorithm called the proximal alternating difference of convex functions (PADCF) algorithm. Moreover, we prove the sequence generated by the PADCF algorithm converges to the critical point under some mild conditions. Numerical experiments are provided for illustrations and comparisons.