Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm

TL;DR

This paper introduces a novel tensor nuclear norm based on the tensor-tensor product to improve the robustness and accuracy of tensor RPCA, with theoretical guarantees and practical applications in image recovery.

Contribution

The paper proposes a new tensor nuclear norm derived from the tensor-tensor product, providing a convex relaxation for tensor rank and enabling exact recovery in TRPCA.

Findings

The new tensor nuclear norm is the convex envelope of tensor average rank.

The proposed TRPCA method guarantees exact recovery under certain conditions.

Numerical experiments demonstrate the effectiveness in image recovery and background modeling.

Abstract

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product). Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.