A novel shrinkage operator for tensor completion with low-tubal-rank approximation

Abstract

The problem of tensor completion (TC) has significant practical significance and a wide range of application backgrounds. Various approaches have been used to solve this problem, including approximating the rank function through its convex envelope, the tensor nuclear norm. However, because of the gap between the rank function and its convex envelope, this method is often unsatisfactory in terms of achieving recovery. As a result, researchers have widely studied explicit non-convex functions that can better approximate the rank function. In this paper, we construct a novel shrinkage operator that functions as the proximal operator of a non-convex function satisfying three critical properties: unbiasedness, sparsity, and continuity. While an explicit analytical expression for the induced non-convex function cannot be obtained, simulation experiments show that it approximates the rank function. By implementing the shrinkage operator in the TC framework, we can show that our iterative sequence converges to the Karush-Kuhn-Tucker (KKT) point. We have discovered that the convergence of the model can be guaranteed as long as certain properties hold for the shrinkage operator. Hence, this result can be extended to a class of shrinkage operators. Extensive experimental results illustrate that our proposed method achieves better recovery performance at the same sampling rate compared to other methods.