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 \(\ell _1\) norm to identify the sparse tensor. For brevity, we refer to the combination of TNF and \(\ell _1\) 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 \(\ell _1\) norm with the ratio of the \(\ell _1\) and Frobenius norm for tensors, the latter denoted as the \(\ell _F\) norm. We refer to the combination of TNF and \(\ell _1/\ell _F\) 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.
