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

TL;DR

This paper introduces a tensor factorization approach for low rank matrix and tensor completion, reformulating the problems to improve efficiency and exploiting low rank structures for better accuracy and speed.

Contribution

It establishes a relationship between matrix rank and tensor tubal rank, and proposes a novel tensor completion model using a two-stage factorization strategy.

Findings

Outperforms state-of-the-art methods in accuracy

Achieves faster computation times

Effectively exploits low rank structures

Abstract

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.