# Weighted Group Sparse Regularized Tensor Decomposition for Hyperspectral Image Denoising
## 2 Notations and Tensor Ring
### Tensor Ring  
In this segment, we present the technique of the tensor ring and the corresponding definitions linked to it. The tensor ring structure is a superior and effective form of decomposition in comparison to other types of tensor decomposition. TR decomposition can be represented as a sequence of cyclic multiplicative third-order tensors with a higher-order tensor. The \(n\)th-tensor is denoted as \(\mathcal{X}\in\mathbb{R}^{l_{1}\times\cdots\times l_{n}}\). The representation of the tensor ring can be decomposed into a series of latent tensors,  
\[\mathcal{X}(i_{1},i_{2},\ldots,i_{n})=T_{\mathcal{H}}\{\mathbf{G }_{1}(i_{1})\mathbf{G}_{2}(i_{2})\ldots\mathbf{G}_{n}(i_{n})\}\] \[=T_{\mathcal{H}}\bigcap_{k=1}^{n}\mathcal{G}_{k}(i_{k})\} \tag{1}\]  
where \(\mathcal{X}(i_{1},i_{2},\ldots,i_{n})\) is the \((i_{1},i_{2},\ldots,i_{n})\)th element of the tensor. \(\mathbf{G}_{k}(i_{k})\) represents the \(i_{k}\)th lateral slice matrix of the latent tensor \(\mathcal{G}_{k}\). It has a size of \(r_{k}\times r_{k+1}\). Any two adjacent latent tensors \(\mathcal{G}_{k}\) and \(\mathcal{G}_{k+1}\) have equivalent dimensions of \(r_{k+1}\) on their corresponding modes. The last latent tensor \(\mathcal{G}_{n}\) has a size of \(r_{n}\times l_{n}\times r_{1}\), i.e., \(r_{n+1}=r_{1}\), which ensures that the product of these matrices is a square matrix. In some cases, the latent tensor \(\mathcal{G}_{k}\) is called the \(k\)th-core. \(r_{k}(k=1,2,\ldots,n)\) is the size of the cores. \(\mathbf{r}=[r_{1},r_{2},\ldots,r_{n}]\) are known as TR ranks. According to (1), the trace of a sequential product of matrices \(\{\mathbf{G}_{k}(i_{k})\}\) is equivalent to \(\mathcal{X}(i_{1},i_{2},\ldots,i_{n})\). In addition, (1) can be rewritten as follows:  
\[\mathcal{X}(i_{1},i_{2},\ldots,i_{n})=\sum_{\alpha_{k},\ldots,\alpha_{n}}^{r_{ 1},\ldots,r_{n}}\prod_{k=1}^{n}G_{k}(\alpha_{k},i_{k},\alpha_{k+1}). \tag{2}\]where \(k\) represents the tensor modes, \(i_{k}\) represents the data dimension index and \(\alpha_{k}\) represents the latent dimension index. For \(\forall k\in\{1,\ldots,n\}\), \(1\leq\alpha_{k}\leq r_{k}\), \(1\leq i_{k}\leq l_{k}\), Equation (2) can also be written as follows:  
\[\mathcal{X}=\sum_{\alpha_{1},\ldots,\alpha_{n}=1}^{r_{1}\ldots r_{n}}\mathbf{g }_{1}(\alpha_{1},\alpha_{2})\circ\mathbf{g}_{2}(\alpha_{2},\alpha_{3})\circ \cdots\circ\mathbf{g}_{n}(\alpha_{n},\alpha_{1}). \tag{3}\]  
where '\(\circ\)' is the outer product of the vector. \(\mathbf{g}_{k}(\alpha_{k},\alpha_{k+1})\in\mathbb{R}^{l_{k}}\) is the \((\alpha_{k},\alpha_{k+1})\) mode-2 fiber of tensor \(\mathcal{X}\).  
Figure 1 graphically represents the tensor ring using a linear tensor network. The order of each tensor is determined by its edges, with every node representing a tensor. The size of each mode is indicated by the number on the corresponding edge. The tensor contraction, also known as the multilinear product operator, occurs when two tensors are connected on a specific mode. For a more comprehensive explanation, please refer to the provided reference [20].  
**Definition 1**.: _(Multilinear Product). For the two adjacent cores of \(TR\) decomposition \(\mathcal{G}_{k}\) and \(\mathcal{G}_{k+1}\), \(\mathcal{G}_{k,k+1}\in\mathbb{R}^{l_{k}\times l_{k}l_{k+1}\times r_{k+1}}\) is the multilinear product of the two cores:_  
\[\mathcal{G}_{k,k+1}(I_{n}(i_{k}-1)+j_{k})=\mathcal{G}_{k}(i_{k})\mathcal{G}_{ k+1}(j_{k}). \tag{4}\]  
_where \(i_{k}=1,\ldots,I_{k},j_{k}=1,\ldots,I_{k+1}\)._  
According to Definition 1, it is established that the tensor \(\mathcal{X}\) can be expressed as the multilinear outcome of a series of core tensors of the third order, known as TR decomposition, i.e.,  
\[\mathcal{X}=\Phi([\mathcal{G}]), \tag{5}\]  
where \(\Phi\) is the reconfiguration operator of TR and \([\mathcal{G}]=\prod\limits_{k=1}^{n}\mathcal{G}_{k}=\{\mathcal{G}_{1}, \mathcal{G}_{2},\ldots,\mathcal{G}_{n}\}\).