Low-M-Rank Tensor Completion and Robust Tensor PCA

TL;DR

This paper introduces the M-rank as a novel, computationally efficient approximation to CP-rank for low-rank tensor completion and robust tensor PCA, outperforming existing methods.

Contribution

It proposes the M-rank and its variants as new tensor ranks, connecting them to CP-rank, and demonstrates their effectiveness in tensor completion and PCA tasks.

Findings

M-rank closely approximates CP-rank

Outperforms existing tensor completion methods

Numerical results validate the approach

Abstract

In this paper, we propose a new approach to solve low-rank tensor completion and robust tensor PCA. Our approach is based on some novel notion of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank. We discuss the connections between these new tensor ranks and the CP-rank and the symmetric CP-rank of an even-order tensor. We show that the M-rank provides a reliable and easy-computable approximation to the CP-rank. As a result, we propose to replace the CP-rank by the M-rank in the low-CP-rank tensor completion and robust tensor PCA. Numerical results suggest that our new approach based on the M-rank outperforms existing methods that are based on low-n-rank, t-SVD and KBR approaches for solving low-rank tensor completion and robust tensor PCA when the underlying tensor has low CP-rank.