Tensor robust principal component analysis via the tensor nuclear over Frobenius norm

TL;DR

This paper introduces a novel tensor robust PCA method using the tensor nuclear over Frobenius norm, demonstrating superior performance in separating low-rank and sparse components in tensors through theoretical analysis and extensive experiments.

Contribution

The paper proposes a new nonconvex tensor nuclear norm approximation for TRPCA, along with an alternative L1/Frobenius norm ratio, and develops an ADMM-based solution with proven convergence.

Findings

Outperforms state-of-the-art TRPCA methods on synthetic and real data

The proposed models effectively separate low-rank and sparse tensors

Theoretical guarantees under incoherence conditions

Abstract

We address the problem of tensor robust principal component analysis (TRPCA), which entails decomposing a given tensor into the sum of a low-rank tensor and a sparse tensor. By leveraging the tensor singular value decomposition (t-SVD), we introduce the ratio of the tensor nuclear norm to the tensor Frobenius norm (TNF) as a nonconvex approximation of the tensor's tubal rank in TRPCA. Additionally, we utilize the traditional L1 norm to identify the sparse tensor. For brevity, we refer to the combination of TNF and L1 as simply TNF. Under a series of incoherence conditions, we prove that a pair of tensors serves as a local minimizer of the proposed TNF-based TRPCA model if one tensor is sufficiently low in rank and the other tensor is sufficiently sparse. In addition, we propose replacing the L1 norm with the ratio of the L1 and Frobenius norm for tensors, the latter denoted as the LF norm. We refer to the combination of TNF and L1/LF as the TNF+ model in short. To solve both TNF and TNF+ models, we employ the alternating direction method of multipliers (ADMM) and prove subsequential convergence under certain conditions. Finally, extensive experiments on synthetic data, real color images, and videos are conducted to demonstrate the superior performance of our proposed models in comparison to state-of-the-art methods in TRPCA.