# Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation
## 3 Notations and Definitions
\begin{split}&\left(\widetilde{\mathbf{F}}_{N_{3}}\otimes \mathbf{I}_{N_{1}}\right)\cdot\mathrm{bcirc}(\mathcal{X})\cdot\left( \widetilde{\mathbf{F}}_{N_{3}}^{*}\otimes\mathbf{I}_{N_{2}}\right)\\ =&\mathrm{bdiag}(\hat{\mathcal{X}})=\begin{bmatrix} \hat{X}_{1}&&\\ &\ddots&\\ &&\hat{X}_{N_{3}}\end{bmatrix},\end{split} \]  
where \(\widetilde{\mathbf{F}}_{N_{3}}=\frac{1}{\sqrt{N_{3}}}\mathbf{F}_{N_{3}}\), \(\mathbf{F}_{N_{3}}\) denotes the \(N_{3}\times N_{3}\) fast Fourier transform matrix, \(\mathbf{F}_{N_{3}}^{*}\) denotes its conjugate, and the notation \(\otimes\) denotes the standard Kronecker product. In addition, \(\mathcal{X}=\mathcal{A}*\mathcal{B}\) if and only if \(\hat{X}_{n_{3}}=\hat{A}_{n_{3}}\hat{B}_{n_{3}}\) for \(1\leq n_{3}\leq N_{3}\) (p.4 of [19]). Since \(\widetilde{\mathbf{F}}_{N_{3}}\otimes\mathbf{I}_{N_{1}},\widetilde{\mathbf{F }}_{N_{3}}\otimes\mathbf{I}_{N_{2}}\) are unitary matrices, we have  
\[\parallel\mathrm{bdiag}(\hat{\mathcal{X}})\parallel_{F}^{2}=\mathrm{tr}\ \mathrm{bcirc}(\mathcal{X})^{\mathsf{T}}\mathrm{bcirc}(\mathcal{X})=N_{3}|| \mathcal{X}||_{F}^{2}.\]  
Then we have \(\parallel\mathcal{X}\parallel_{F}=\frac{1}{\sqrt{N_{3}}}\parallel\mathrm{bdiag }(\hat{\mathcal{X}})\parallel_{F}\). **Definition 4** (Tensor multi-rank p. 10 of [37]). : _For tensor \(\mathcal{X}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}\) with \(N_{1}\times N_{2}\) frontal slices \(X_{1},\ldots,X_{N_{3}}\), the multi-rank of \(\mathcal{X}\) is a \(N_{3}\)-vector \([\text{rank}(\hat{X}_{1}),\ldots,\text{rank}(\hat{X}_{N_{3}})]\), consisting of the ranks of all the \(\hat{X}_{1},\ldots,\hat{X}_{N_{3}}\)._  
Please note that the matrix SVD [39] can be performed on each frontal slice of \(\hat{\mathcal{X}}\), The matrix \(\hat{X}_{n_{3}}\) has the following Singular Value Decomposition (SVD) \(\hat{X}_{n_{3}}=\hat{U}_{n_{3}}\hat{\Sigma}_{n_{3}}\hat{V}_{n_{3}}^{\mathsf{T}}\), \(\hat{U}_{n_{3}}\in\mathbb{O}(N_{1})\), \(\hat{V}_{n_{3}}\in\mathbb{O}(N_{2})\) (here \(\mathbb{O}(N_{i})\) stands for the set of orthogonal \(N_{i}\)-by-\(N_{i}\) matrices) and \(\hat{\Sigma}_{n_{3}}\) is an \(N_{1}\times N_{2}\) diagonal matrix with diagonal entries \(\sigma_{1}(\hat{X}_{n_{3}}),\ldots,\sigma_{\min\{N_{1},N_{2}\}}(\hat{X}_{n_{3}})\) in descending order. **Definition 5** (t-SVD, Theorem 4.1 in [13]). : _The tensor Singular Value Decomposition (t-SVD) of \(\mathcal{X}\) is given by \(\mathcal{X}=\mathcal{U}*\Sigma*\mathcal{V}^{\mathsf{T}}\), 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. \(\Sigma\) is an f-diagonal tensor of size \(N_{1}\times N_{2}\times N_{3}\), and \(*\) denotes the t-product._  
**Definition 6** (Tensor tubal rank III.B of [21]). : _The tensor tubal rank is defined as the number of nonzero singular tubes of f-diagonal tensor \(\Sigma\), which is the maximum of the number of nonzero \(\{\hat{\Sigma}_{n,n,n_{3}},\ 1\leq n\leq\min(N_{1},N_{2})\}\) for \(1\leq n_{3}\leq N_{3}\)._  
**Definition 7** (Tensor nuclear norm III.B of [21]). : _The nuclear 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_  
\[\parallel\mathcal{X}\parallel_{\text{TNN}}=\frac{1}{N_{3}}\sum_{n_{3}=1}^{N_{ 3}}||\hat{X}_{n_{3}}||_{*},\]  
_which is equal to_  
\[\sum_{n=1}^{\min(N_{1},N_{2})}\sum_{n_{3}=1}^{N_{3}}\frac{1}{N_{3}}\hat{ \Sigma}_{n,n,n_{3}}.\]  
_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._  
In this paper, to overcome the disadvantages of the tensor nuclear norm, we propose smooth but more effective, though being non-convex, t-SVD-based non-convex rank approximation to the data sets represented by 3 rd-order tensor. This will make the larger singular values be shrunk less in order to preserve their characteristics.