Exact Recovery of Tensor Robust Principal Component Analysis under Linear Transforms

TL;DR

This paper introduces a new tensor robust PCA method using linear transforms, providing theoretical guarantees for exact recovery of low-rank and sparse components, and demonstrating superior performance in image recovery tasks.

Contribution

The work develops a generalized TRPCA model based on linear transforms, extending previous Fourier-based methods with theoretical recovery guarantees.

Findings

The convex program exactly recovers low-rank and sparse components under certain conditions.

The model generalizes existing matrix and Fourier-based TRPCA methods.

Numerical experiments confirm the effectiveness and superiority of the proposed approach.

Abstract

This work studies 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 motivated by the recently proposed linear transforms based tensor-tensor product and tensor SVD. We define a new transforms depended tensor rank and the corresponding tensor nuclear norm. Then we solve the TRPCA problem by convex optimization whose objective is a weighted combination of the new tensor nuclear norm and the $\ell_1$-norm. In theory, we show that under certain incoherence conditions, the convex program exactly recovers the underlying low-rank and sparse components. It is of great interest that our new TRPCA model generalizes existing works. In particular, if the studied tensor reduces to a matrix, our TRPCA model reduces to the known matrix RPCA. Our new TRPCA which is allowed to use general linear transforms can be regarded as an extension of our former TRPCA work which uses the discrete Fourier transform. But their proof of the recovery guarantee is different. Numerical experiments verify our results and the application on image recovery demonstrates the superiority of our method.