# Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation
## 3 Notations and Definitions
Moreover, under our framework, we establish explicit solutions to the corresponding non-convex tensor norm minimization problems that is not available in current literature. **Definition 8** (Gamma quasi-norm for matrix [26]). : _Let \(X\in\mathbb{R}^{m\times n}\) and the singular values of \(X\), denoted by \(\sigma_{1}(X),\ldots,\sigma_{\min\{m,n\}}(X)\). We could introduce the Gamma norm of \(X\), which is defined as \(||X||_{\gamma}=\sum_{i=1}^{\min m,n}\frac{(1+\gamma)\sigma_{i}(X)}{\gamma+ \sigma_{i}(X)},\gamma>0\)._  
**Definition 9** (t-Gamma tensor quasi-norm). : _The Gamma tensor quasi-norm for a given 3 rd-order tensor \(\mathcal{X}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}\) is defined to be_  
\[||\mathcal{X}||_{t\sim\gamma}=\frac{1}{N_{3}}\sum_{n_{3}=1}^{N_{3}}||\hat{X}_{ n_{3}}||_{\gamma},\]  
_which is equal to_  
\[\frac{1}{N_{3}}\sum_{n_{3}=1}^{N_{3}}\left(\sum_{n=1}^{\min\{N_{1},N_{2}\}} \frac{(1+\gamma)\hat{\Sigma}_{n,n,n_{3}}}{\gamma+\hat{\Sigma}_{n,n,n_{3}}} \right).\]  
_Here \(\hat{X}_{n_{3}}\) is the \(n_{3}\)-th frontal slice of \(\hat{\mathcal{X}}=\text{fft}(\mathcal{X},[\ ],3)=[\hat{X}_{1}|\cdots|\hat{X}_{N_{3}}]\in \mathbb{R}^{N_{1}\times N_{2}\times N_{3}}\) obtained by applying FFT on \(\hat{\mathcal{X}}\) along the third mode._  
**Remark 1**. : _It is not hard to see the t-Gamma tensor quasi-norm is unitarily invariant, that is \(||\mathcal{X}||_{t\sim\gamma}=||\mathcal{U}*\mathcal{X}*\mathcal{V}^{\mathsf{ T}}||_{t\sim\gamma}\), where \(\mathcal{U}\) and \(\mathcal{V}\) are orthogonal tensors of size \(N_{1}\times N_{1}\times N_{3}\) and \(N_{2}\times N_{2}\times N_{3}\) respectively. In addition,_  
\[\lim_{\gamma\to 0}||\mathcal{X}||_{t\sim\gamma}=\text{tubal rank of }\mathcal{X};\ \lim_{\gamma\to\infty}||\mathcal{X}||_{t\sim\gamma}=||\mathcal{X}||_{\text{TNN}}.\]  
**Lemma 1** (Theorem 1 of [26]). : _Let \(A=U\Sigma_{A}V^{T}\) be the SVD of \(A\in\mathbb{R}^{m\times n}\) and \(\Sigma_{A}=\text{diag}(\sigma_{A})\) are singular values of \(A\). Let \(F(Z)=f\circ\sigma(Z)\) be a unitary invariant function and \(\mu>0\). Then an optimal solution to the following problem_  
\[\min_{Z}F(Z)+\frac{\mu}{2}\left\|Z-A\right\|_{F}^{2}, \tag{3}\]  
_is \(Z^{*}=U\Sigma_{Z}^{*}V^{T}\), where \(\Sigma_{Z}^{*}=\text{diag}(\sigma^{*})\) and \(\sigma^{*}=\text{prox}_{f,\mu}(\sigma_{A})\). Here \(\text{prox}_{f,\mu}(\sigma_{A})\) is the Moreau-Yosida operator [40], defined as_  
\[\text{prox}_{f,\mu}(\sigma_{A}):=\text{argmin}_{\sigma\geq 0}f(\sigma)+\frac{ \mu}{2}\|\sigma-\sigma_{A}\|_{2}^{2}. \tag{4}\]  
Now let us consider the tensor t-Gamma quasi-norm minimization problem:  
**Theorem 1** (t-Gamma tensor rank minimization). : _Let us consider a 3 rd-order tensor \(\mathcal{X}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}\), which has t-SVD \(\mathcal{X}=\mathcal{U}*\Sigma*\mathcal{V}^{\text{T}}\) as in Definition 5. If \(\mathcal{Z}^{*}\) minimizes_  
\[\frac{1}{2}\|\mathcal{X}-\mathcal{Z}\|_{F}^{2}+\tau\parallel\mathcal{Z} \parallel_{t\sim},\]  
_then_  
\[\mathcal{Z}^{*}=\mathcal{U}*\Omega_{\mathcal{Z}}*\mathcal{V}^{\text{T}},\]  
_where \(\Omega_{\mathcal{Z}}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}\) is a f-diagonal tensor. For \(1\leq n\leq\min(N_{1},N_{2}),1\leq n_{3}\leq N_{3}\), let \(\hat{\Sigma}=[\hat{\Sigma}_{n,n,n_{3}}]=\text{fft}(\Sigma,[\,],3),\ \hat{\Omega}_{ \mathcal{Z}}=[\hat{\Omega}_{n,n,n_{3}}]=\text{fft}(\Omega,[\,],3)\), then \(\hat{\Omega}_{n,n,n_{3}}\) is the limit point of Fixed Point Iteration_  
\[\parallel\mathcal{X}\parallel_{\text{TNN}}=\frac{1}{N_{3}}\sum_{n_{3}=1}^{N_{ 3}}||\hat{X}_{n_{3}}||_{*},\]0  
**Remark 2**. : _Theorem 1 provides the connection of the optimal solution between the original domain and the Fourier domain. More importantly, it shows that the minimization problem under our framework is tractable. After the original tensor \(\mathcal{X}\) is transformed to \(\hat{\mathcal{X}}\) in the Fourier domain, an equivalent minimization problem could be solved by using Lemma 1 for each slice \(\hat{X}_{n_{3}},1\leq n_{3}\leq N_{3}\) in the Fourier domain.