# Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation
## 1 Introduction
Then the minimization of \(\|L\|_{*}\) will attempt to minimize \(\sigma_{1}\) that leads to over-penalizes \(\sigma_{1}\), and consequently it generates undesirable solution \(L\) with small \(\sigma_{1}\). Recall that our goal is to find a low-rank \(L\), whose largest singular value is not necessarily to be small. In addition, it is worth mentioning that in practice the matrix RPCA problem is very sensitive to small perturbations and this problem can be addressed by relaxing the equality constraints. The authors in [22; 23] propose an alternative optimization approach that appears to be suitable to deal with robustness issues in the \"Sparse Plus Low-rank\" decomposition problem. It has long been noticed that the nuclear norm is not always satisfactory [24], since it over-penalizes large singular values, and consequently may only find a much biased solution. Since the the nuclear norm is the tightest convex relaxation of the rank function [25], a further improvement under the convex setting in general is limited, if it is still possible, in particular when their gap is not very small. In order to alleviate the above problem, the Gamma quasi-norm for _matrices_ is introduced in [26] in RPCA, which results in a tighter approximation of the rank function. For \(L\in\mathbb{R}^{m\times n}\) and the singular values of \(L\), denoted by \(\sigma_{1}(L),\ldots,\sigma_{\min\{m,n\}}(L)\), the Gamma quasi-norm of \(L\) is defined as  
\[||L||_{\gamma}=\sum_{i=1}^{\min\{m,n\}}\frac{(1+\gamma)\sigma_{i}(L)}{\gamma+ \sigma_{i}(L)},\gamma>0.\]  
It is not difficult to see  
\[\frac{(1+\gamma)\sigma_{i}(L)}{\gamma+\sigma_{i}(L)}<\frac{1+\gamma}{\gamma+M }\sigma_{i}(L),\]when \(\sigma_{i}(L)>M>0\) and \(||L||_{\gamma}\leq(1+\gamma)\) rank\((L)\). So it overcomes the imbalanced penalization by different singular values in convex nuclear norm. In addition, when \(\gamma\) goes to 0, the limit of the Gamma quasi-norm is exactly the matrix rank. The Gamma quasi-norm has nice algebraic properties, i.e., unitarily invariant and positive definiteness. Thanks to the unitarily invariance [27], the Gamma quasi-norm minimization problem for \(A\in\mathbb{R}^{m\times n}\):  
\[\min_{Z\in\mathbb{R}^{m\times n}}||Z||_{\gamma}+\frac{\mu}{2}\,||Z-A||_{F}^{2}\]  
can be solved very effectively (Theorem 1 of [26]). Moreover, experimental results in [26] show that Gamma quasi-norma-based RPCA outperforms several other convex/non-convex norma-based RPCA in both accuracy and efficiency. In this paper, inspired by the success of the non-convex rank approximation [26], we proposed tensor Gamma quasi-norm via t-SVD to seek a closer approximation and to alleviate the above-mentioned limitations resulted from the tensor nuclear norm, which has not been used for TRPCA yet in literature. The main contributions of this paper are illustrated as follows:  
1. We develop new non-convex tensor rank approximation functions that can be used as desirable regularization for the optimization process, which appears to be more effective than current existing approaches in literature. 2. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the equivalent optimization problem in the Fourier domain, which is also suitable for general TRPCA problem. In addition, we provided the convergent analysis of the augmented Lagrange multiplier based optimization algorithm. 3. Extensive evaluation of the proposed approach on several benchmark data sets has been conducted, and detailed comparison with most recent approaches are provided. Experimental results demonstrate that our proposed approach yields superior performance for image recovery and video background modeling. The rest of the paper is organized as follows. In Section 2, we roughly describe some related works. Then Section 3 provides factorization strategies for 3 rd-order tensors and its relationship to the corresponding tensor approximation problem. And then we present the main results on t-Gamma quasi-norm of 3 rd-order tensors.