## 3 Preliminaries
### Introduction to Tensors
This operator reshapes tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) into a block circulant matrix of size \(G\in\mathbb{R}^{n_{1}n_{3}\times n_{2}n_{3}}\). The block circulant matricization operator arranges the frontal slices of tensor A in a circulant manner, thereby preserving the temporal relationships across the frontal slices and the multivariate correlations within the frontal slices. \[circ(\mathcal{G})= \begin{bmatrix}G^{(1)}&G^{(n_{3})}&\cdots&G^{(2)}\\ G^{(2)}&G^{(1)}&\cdots&G^{(3)}\\ \vdots&\vdots&\vdots&\vdots\\ G^{(n_{3})}&G^{(n_{3}-1)}&\cdots&G^{(1)}\end{bmatrix}  \tag{3}\]  
Within the tensor-SVD framework, the tensor product is conceptualized as an extension of the matrix product. **Definition 1**. : _The **tensor product**[21] involves two tensors, \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) and \(\mathcal{Y}\in\mathbb{R}^{n_{1}\times n_{4}\times n_{3}}\). The \(t\)-product, represented by \(\mathcal{G}*\mathcal{Y}\), is defined as a tensor of size \(n_{1}\times n_{4}\times n_{3}\). This is given by_  
\[\mathcal{G}*\mathcal{Y}=fold(circ(\mathcal{G})\cdot(unfold(\mathcal{Y})) \tag{4}\]  
**Definition 2**. : _A **conjugate transpose**, as detailed in [21], involves a tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\). The conjugate transpose of this tensor, denoted by \(\mathcal{G}^{*}\in\mathbb{R}^{n_{2}\times n_{1}\times n_{3}}\), is obtained by taking the conjugate transpose of each frontal slice and subsequently reversing the order of the transposed frontal slices from 2 to \(n_{3}\)._  
**Definition 3**. : _An **identity tensor**[21] is a tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\). This tensor is characterized by its first frontal slice being an \(n\times n\) identity matrix, whereas all other frontal slices consist entirely of zeros._  
Figure 2: Fibers of third-order tensor \(\mathcal{G}\): (**a**) tube fibers; (**b**) column fibers; (**c**) row fibers. **Definition 4**. : _A **block diagonal matrix**[41] with \(\hat{\mathcal{G}}=fft(\mathcal{G})\) represents the tensor obtained by executing a fast Fourier transform (FFT) along the third dimension of the tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\). The block diagonal matrix of \(\hat{\mathcal{G}}\) is defined as follows:_  
\[\hat{\mathcal{G}}=\text{blockdiag}(\hat{\mathcal{G}})= \begin{bmatrix}G^{(1)}&&& 0\\ &G^{(2)}&&\\ &&\ddots&\\ 0&&&G^{(n_{3})}\end{bmatrix}  \tag{5}\]  
**Definition 5**. : _An **orthogonal tensor**[21] is a tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{1}\times n_{3}}\). This tensor is considered orthogonal if it fulfills the following condition:_  
\[\mathcal{G}*\mathcal{G}^{T}=\mathcal{G}^{T}*\mathcal{G}=\mathcal{I} \tag{6}\]  
**Definition 6**. : \(f-diagonal\) **tensor**[21] is a tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{1}\times n_{3}}\). This tensor is considered orthogonal if it satisfies the following conditions. **Theorem 1**. : _Tensor Singular Value Decomposition (t-SVD) [21] is a method of decomposing a third-order tensor. It breaks down a tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) into three components: a core tensor \(\mathcal{D}\in\mathbb{R}^{n_{1}\times n_{1}\times n_{3}}\), an orthogonal tensor on the right \(\mathcal{U}\in\mathbb{R}^{n_{1}\times n_{1}\times n_{3}}\), and an orthogonal tensor on the left \(\mathcal{V}\in\mathbb{R}^{n_{2}\times n_{2}\times n_{3}}\). The decomposition is expressed as follows:_  
\[\mathcal{G}=\mathcal{U}*\mathcal{D}*\mathcal{V}^{T} \tag{7}\]  
_In the t-SVD process for a third-order tensor, as illustrated in Algorithm 1, tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) is decomposed into three distinct tensors. \(\mathcal{U}\) and \(\mathcal{V}\) are orthogonal tensors, whereas \(\mathcal{D}\) is an \(f-diagonal\) core tensor.