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Low rank matrix and tensor completion problems are to recover the incompletetwo and higher order data by using their low rank structures. The essentialproblem in the matrix and tensor completion problems is how to improve theefficiency. To this end, we first establish the relationship between matrixrank and tensor tubal rank, and then reformulate matrix completion problem as atensor completion problem. For the reformulated tensor completion problem, weadopt a two-stage strategy based on tensor factorization algorithm. In thisway, a matrix completion problem of big size can be solved via some matrixcomputations of smaller sizes. For a third order tensor completion problem, tofully exploit the low rank structures, we introduce the double tubal rank whichcombines the tubal rank and the rank of the mode-3 unfolding matrix. For themode-3 unfolding matrix rank, we follow the idea of matrix completion. Based onthis, we establish a novel model and modify the tensor factorization basedalgorithm for third order tensor completion. Extensive numerical experimentsdemonstrate that the proposed methods outperform state-of-the-art methods interms of both accuracy and running time.

Tensor factorization based method for low rank matrix completion and its application on tensor completion