Tensor Completion via Generalized Tensor Tubal Rank Minimization Using General Unfolding

Abstract

This letter addresses the problem of tensor completion. The properties of the tensor tubal rank (TTR) and tensor Kronecker rank are first discussed, and then a novel generalized tubal Kronecker decomposition together with a new tensor rank referred to as generalized tensor tubal rank (GTTR) are defined. It is shown that the GTTR is suitable for revealing both the Kronecker and tubal structures of a tensor. The general tensor completion idea is then presented following the procedure of alternate projection between tensor rank minimization and Frobenius-norm optimization. Furthermore, the GTTR minimization is relaxed to the problem of generalized tensor nuclear norm (TNN) minimization, and two solutions are derived. The first one is based on the idea of combining all generalized TNNs as a weighted sum, while the second one employs the alternate cancelation scheme. Experiments are also carried out using both simulated data and real datasets for comparison of the proposed and the state-of-the-art approaches.