# Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation
## 2 Related Works  
Recently, many authors observed the limitations of t-SVD-based TRPCA [15; 19]. It was pointed out in [28] that the classical TRPCA methods based on t-SVD fails to utilize the low rank structure in the third mode. To further exploit the low-rank structures in multiway data, they defined a new TNN that extends TNN with core matrix and proposed a creative algorithm to deal with TRPCA problems. In addition, the authors of [29] presented that there are some weak points in the existing low-rank approximation approaches, for example, people need to predefine the rank values or fail to consider local information of data (e.g., spatial or spectral smooth structure). So they proposed a novel TNN-based low-rank approximation with total variation regularization, which uses the TNN to encode the global low-rank prior of tensor data and the total variation regularization to preserve the spatial-spectral continuity in a unified framework.  
It has been noticed by others that TNN is not always satisfactory. For instance, to seek a closer approximation of tensor rank, [PERSON] et al. in [30] introduced t-Schatten-\(p\) tensor quasi-norm to replace the TNN. The t-Schatten-\(p\) norm is convex when \(p\geq 1\), but non-convex when \(0<p<1\). In addition, they proved multi-Schatten-\(p\) norm surrogate theorem to convert the non-convext-Schatten-\(p\) tensor quasi-norm for \(0<p<1\) into a weighted sum of convex functions. Then based on the multi-Schatten-\(p\) norm surrogate, they proposed a novel convex tensor RPCA model.  
Besides above, a recent paper [31] developed randomized algorithms for tensor low-rank approximation and decomposition, together with tensor multiplication. Their proposed tubal focused algorithms employ a small number of lateral and/or horizontal slices of the underlying 3 rd order tensor, that come with relative error guarantees for the quality of the obtained solutions.  
Besides the above-mentioned methods based on t-SVD, there are many other TRPCA methods mainly applied CP (CANDECOMP/PARAFAC) or Tucker decompositions methods in their optimization models. We need to mention that recently [PERSON] et al. [32] studied TRPCA using a non-convex formulation of the CP based tensor atomic-norm and identified a class of local minima of this non-convex program that are globally optimal.  
Presently, background subtraction become more and more important for visual surveillance systems and several subspace learning algorithms based on matrix and tensor tools have been used to perform the background modeling of the scenes. Recently, [PERSON] et al. [33] developed an incremental tensor subspace learning that uses only a small part of the entire data and updates the low-rank model incrementally when new data arrives and in [34] the authors proposed an online stochastic framework for tensor decomposition of multispectral video sequences. In particular, it is worth mentioning that [PERSON] et al. [35] gave a systematic and complete review of the \"low-rank plus sparse matrix decomposition\" framework of separating moving objects from the background.  
However, all those above mentioned t-SVD-based TRPCA algorithms solve the TRPCA optimization problem under convex regularization. In addition, most analyses of ADMM algorithms are in the convex setting [36]. In our work, we explore the TRPCA method based on the non-convex t-Gamma quasi-norm of 3 rd-order tensors. Due to the non-convex nature of our TRPCA model, the convergent analysis of algorithm is difficult to justify. In addition, the analysis of non-convex augmented Lagrangians is not easy, but this compels us to make efforts of clarity and pedagogy. Also it should be noted that there are few works on t-SVD-based TRPCA achieve background modeling. Thus experiments are provided in our paper on two computer vision applications, which are image recovery and video background modeling.