T-product factorization based method for matrix and tensor completion problems

Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data of low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency. For a matrix completion problem, we establish a relationship between matrix rank and tensor tubal rank, and reformulate matrix completion problem as a third order tensor completion problem. For the reformulated tensor completion problem, we adopt a two-stage strategy based on tensor factorization algorithm. In this way, a matrix completion problem of big size can be solved via some matrix computations of smaller sizes. For a third order tensor completion problem, to fully exploit the low rank structures, we introduce the double tubal rank which combines the tubal rank of two tensors, original tensor and the reshaped tensor of the mode-3 unfolding matrix of original tensor. Based on this, we propose a reweighted tensor factorization algorithm for third order tensor completion. Extensive numerical experiments demonstrate that the proposed methods outperform state-of-the-art methods in terms of both accuracy and running time.