# Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation
## 4 Tensor Robust Principal Component Analysis with Non-Convex Regularization  
A TRPCA model using non-convex regularization can be formulated as  
\[ \begin{split}\min_{\mathcal{L},\mathcal{E}}&\| \mathcal{L}\|_{t\gamma}+\lambda\|\mathcal{E}\|_{\ell_{1}}\\ \text{s.t.}&\mathcal{X}=\mathcal{L}+\mathcal{E}. \end{split}  \tag{5}\]  
Its augmented Lagrangian function is \(L_{\beta}(\mathcal{L},\mathcal{E},\mathcal{M})=\|\mathcal{L}\|_{t\gamma}+ \lambda\|\mathcal{E}\|_{\ell_{1}}+\frac{\beta}{2}\|\mathcal{X}-\mathcal{L}- \mathcal{E}-\frac{M}{\beta}\|_{F}^{2}+\mathcal{C}\), where \(\mathcal{M}\) is the Lagrangian multiplier, \(\beta\) is the penalty parameter for the violation of the linear constraints, and \(\mathcal{C}\) is a constant. Then, the problem \(\arg\min_{\mathcal{L},\mathcal{E},\mathcal{M}}L_{\beta}(\mathcal{L},\mathcal{E },\mathcal{M})\) can be updated as:  
\[\left\{ \begin{array}{l}\mathcal{L}_{k+1}=\operatorname*{arg\,min}_{\mathcal{ L}}\|\mathcal{X}-\mathcal{E}_{k}-\frac{\mathcal{M}_{k}}{\beta}-\mathcal{L}\|_{F}^{2} +\frac{2}{\beta}\|\mathcal{L}\|_{t\gamma},\\ \mathcal{E}_{k+1}=\mathcal{S}_{\frac{\beta}{\beta}}\left(\mathcal{X}- \mathcal{L}_{k+1}+\frac{\mathcal{M}^{k}}{\beta}\right),\\ \mathcal{M}_{k+1}=\mathcal{M}_{k}+\beta(\mathcal{X}-\mathcal{L}_{k+1}- \mathcal{E}_{k+1}),\end{array} \right. \tag{6}\]  
where the tensor non-negative **soft-thresholding operator**\(\mathcal{E}_{v}(\cdot)\)[43] is defined as  
\[\mathcal{S}_{v}(\mathcal{B})=\mathcal{B}\]  
with  
\[\bar{b}_{n_{1}n_{2}n_{3}}= \begin{cases}sgn(b_{n_{1}n_{2}n_{3}})(|b_{n_{1}n_{2} n_{3}}|-v),&|b_{n_{1}n_{2}n_{3}}|>v,\\ 0,&|b_{n_{1}n_{2}n_{3}}|\leq v.\end{cases} \]Algorithm 1 shows the pseudocode for the proposed non-convex tensor robust component analysis method. ```
0: The observed tensor \(\mathcal{X}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}\), the set of index of observed entries \(\Omega\), parameter \(\lambda\), stopping criterion \(\epsilon\). Initialize:\(\mathcal{L}_{\omega>0}^{0}=\mathcal{X}_{\omega>0}\), \(\mathcal{L}_{\Omega^{\mathrm{C}}}^{0}=\mathrm{rand}(N_{1}\times N_{2}\times N_{ 3})_{\Omega^{\mathrm{C}}}\), \(\mathcal{E}^{0}=\mathcal{M}^{0}=\mathrm{zeros}(N_{1}\times N_{2}\times N_{3})\), \(\beta=10^{-4}\), \(\beta_{max}=10^{10},\eta=1.2\). 1:while not converged do
2:\(\mathcal{L}_{k+1}\leftarrow\mathcal{U}_{\mathcal{X}-\mathcal{E}_{k}-\frac{ \mathcal{M}_{k}}{\beta_{k}}}*\Omega_{\mathcal{L}}*\mathcal{V}^{\mathrm{T}}_{ \mathcal{X}-\mathcal{E}_{k}-\frac{\mathcal{M}_{k}}{\beta_{k}}}\) using the local minimum in Theorem 1 with \(\tau=\frac{1}{\beta_{k}}\)
3:\(\mathcal{E}_{k+1}\leftarrow\mathcal{S}_{\frac{\lambda}{\beta_{k}}}\left( \mathcal{X}-\mathcal{L}_{k+1}+\frac{\mathcal{M}_{k}}{\beta_{k}}\right)\)
4:\(\mathcal{M}_{k+1}\leftarrow\mathcal{M}_{k}+\beta_{k}(\mathcal{X}-\mathcal{L} _{k+1}-\mathcal{E}_{k+1})\)
5:\(\beta_{k+1}\leftarrow\min\{\beta_{k}\eta,\beta_{max}\}\)
6: Check the convergence conditions \(\|\mathcal{L}_{k+1}-\mathcal{L}_{k}\|_{\infty}\leq\epsilon\), \(\|\mathcal{E}_{k+1}-\mathcal{E}_{k}\|_{\infty}\leq\epsilon\), \(\|\mathcal{L}_{k+1}+\mathcal{E}_{k+1}-\mathcal{X}\|_{\infty}\leq\epsilon\)
7:endwhile
```  
**Output:** The low rank tensor \(\mathcal{L}\) and the sparse tensor \(\mathcal{E}\)  
**Theorem 2** (Convergent Analysis). : _Suppose that \(\mathcal{P}_{k}=\{\mathcal{L}_{k},\mathcal{E}_{k},\mathcal{M}_{k}\}\) is generated from the proposed Algorithm 1, then \(\{\mathcal{P}_{k}\}\) is bounded. In addition, the accumulation point \(\{\mathcal{L}^{*},\mathcal{E}^{*},\mathcal{M}^{*}\}\) of \(\{\mathcal{P}_{k}\}_{k=1}^{+\infty}\) is a KKT stationary point, which satisfy the KKT conditions \(\mathcal{M}^{*}\in\partial||\mathcal{L}||_{t\text{-}\gamma}\), \(\mathcal{L}^{*}+\mathcal{E}^{*}=\mathcal{X}\), \(\mathcal{M}^{*}\in\partial||\mathcal{E}||_{\ell_{1}}\)._  
**Proof of Theorem 2.