Tensor robust principal component analysis via the tensor nuclear over Frobenius norm
Huiwen Zheng, Yifei Lou, Guoliang Tian, Chao Wang
TL;DR
This work introduces two nonconvex TRPCA models based on the tensor nuclear norm over the Frobenius norm (TNF) to enforce low tubal rank in third-order tensors, paired with either the classical L1 sparsity ($\mathcal{E}$) or the L1/Frobenius sparsity ratio (LF) for robustness. The TNF model uses ADMM with a t-SVT update for the low-rank term and closed-form soft-thresholding for the sparse part, achieving subsequential convergence to a stationary point under incoherence and sampling assumptions; the TNF+ variant extends this with an LF sparsity term and a two-variable ADMM scheme, with convergence guarantees under analogous conditions. Theoretical recovery guarantees (for TNF) are established under tensor incoherence and random sparse support, ensuring local optimality of the true decomposition with high probability as tensor dimensions grow. Empirically, TNF and TNF+ outperform state-of-the-art TRPCA methods on synthetic data, color-image denoising, and video-background modeling, demonstrating superior recovery accuracy (PSNR/SSIM) and competitive runtimes. Overall, the paper provides a principled nonconvex surrogate for tensor rank, practical ADMM algorithms with convergence guarantees, and strong empirical validation for robust tensor decomposition in vision and signal-processing tasks.
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.