Low-Rank Tensor Completion Based on Log-Det Rank Approximation and Matrix Factorization

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Rank evaluation plays a key role in low-rank tensor completion and tensor nuclear norm is often used as a substitute of rank in the optimization due to its convex property. However, this replacement often incurs unexpected errors, and since singular value decomposition is frequently involved, the computation cost of the norm is high, especially in handling large scale matrices from the mode-n unfolding of a tensor. This paper presents a novel tensor completion method, in which a non-convex logDet function is utilized to approximate the rank and a matrix factorization is adopted to reduce the evaluation cost of the function. The study shows that the logDet function is a much tighter rank approximation than the nuclear norm and the matrix factorization can significantly reduce the size of matrix that needs to be evaluated. In the implementation of the method, alternating direction method of multipliers is employed to obtain the optimal tensor completion. Several experiments are carried out to validate the method and the results show that the proposed method is effective.