  Low rank matrix and tensor completion problems are to recover the incomplete
two and higher order data by using their low rank structures. The essential
problem in the matrix and tensor completion problems is how to improve the
efficiency. To this end, we first establish the relationship between matrix
rank and tensor tubal rank, and then reformulate matrix completion problem as a
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 and the rank of the mode-3 unfolding matrix. For the
mode-3 unfolding matrix rank, we follow the idea of matrix completion. Based on
this, we establish a novel model and modify the tensor factorization based
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.
