## 2 Background  
### Tensor Operations  
In this section, the tensor operations that are used throughout this paper are presented. Suppose we have a 3-mode tensor, \(\mathcal{A}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\), the basic tensor operations can be then listed as follows:  
* **Block circulant matrix**, \(bcirc(\mathcal{A})\): Converts the tensor \(\mathcal{A}\) to a matrix by merging all frontal slices vertically and horizontally, which results in a matrix of the dimension \(n_{3}n_{1}\times n_{3}n_{2}\), defined as: \[bcirc(\mathcal{A})= \begin{bmatrix}\mathcal{A}^{(1)}&\mathcal{A}^{(n_{3})}& \ldots&\mathcal{A}^{(2)}\\ \mathcal{A}^{(2)}&\mathcal{A}^{(1)}&\ldots&\mathcal{A}^{(3)}\\ \vdots&\vdots&\ddots&\vdots\\ \mathcal{A}^{(n_{3})}&\mathcal{A}^{(n_{3}-1)}&\ldots&\mathcal{A}^{(1)}\\ \end{bmatrix} ,\] (1) where \(\mathcal{A}^{(i)}\) represents the \(i^{th}\) frontal slice of \(\mathcal{A}\). * **Block vectorizing**, \(bvec(\mathcal{A})\): Converts the tensor \(\mathcal{A}\) to a block matrix by aggregating all frontal slices vertically and this results in a matrix of the dimension \(n_{3}n_{1}\times n_{3}\) defined as follows: \[bvec(\mathcal{A})= \begin{bmatrix}\mathcal{A}^{(1)}\\ \mathcal{A}^{(2)}\\ \vdots\\ \mathcal{A}^{(n_{3})}\\ \end{bmatrix} .\] (2) Furthermore, the opposite operation of the block vectorizing that transforms the aggregated matrix into a tensor can be defined as: \[bvfold(bvec(\mathcal{A}))=\mathcal{A}.\] (3)
* **Block diagonal matrix**, \(bdiag(\mathcal{A})\): Converts the tensor \(\mathcal{A}\) to a block diagonal matrix. This is executed by stacking all frontal slices on the diagonal of the matrix, leading to a matrix of the dimension \(n_{3}n_{1}\times n_{3}n_{2}\) that can be defined as follows: \[bdiag(\mathcal{A})= \begin{bmatrix}\mathcal{A}^{(1)}&&\\ &\ddots&\\ &&\mathcal{A}^{(n_{3})}\\ \end{bmatrix} .\] (4) \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\mathcal{A}\), \(\mathbf{A}\), \(\mathbf{a}\) & Tensor, matrix, vector \\ \(\|\mathbf{A}\|_{F}\) & Frobenius norm of the matrix \(\mathbf{A}\), \\ \(\|\mathcal{A}\|_{F}^{2}\) & Frobenius norm of the tensor \(\mathcal{A}\), \(\|\mathcal{A}\|_{F}^{2}\sum_{i,j,k}a_{ijk}^{2}\) \\ \(\|\mathbf{A}\|_{*}\) & Nuclear norm of \(\mathbf{A}\), \(\|\mathbf{A}\|_{*}=\sum_{i}\sigma_{i}(\mathbf{A})\) \\ \(\sigma_{i}(\mathbf{A})\) & \(i^{th}\) singular value of \(\mathbf{A}\) \\ \(\mathcal{A}(:,:,:)\) & \(j^{th}\) horizontal slice of \(\mathcal{A}\) \\ \(\mathcal{A}(:,:,:)\) & \(j^{th}\) lateral slice of \(\mathcal{A}\) \\ \(\mathcal{A}(:,:,j)\) & \(j^{th}\) frontal slice of \(\mathcal{A}\) \\ \(\mathcal{A}(0)\) & \(j^{th}\) frontal slice of \(\mathcal{A}\) \\ \(\mathcal{A}(:,:,i,j)\) & Mode-1 fiber of \(\mathcal{A}\) \\ \(\mathcal{A}(i,:,j)\) & Mode-2 fiber of \(\mathcal{A}\) \\ \(\mathcal{A}(i,j,:)\) & Mode-3 fiber of \(\mathcal{A}\) \\ \(\mathbf{A}_{+}\) & The positive values of matrix \(\mathbf{A}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of notations. On the other hand, the block diagonal opposite operation \(bdfold(bdiag(\mathcal{A}))=\mathcal{A}\) recovers the tensor from the block diagonal matrix. * **Tensor Product (t-Product)**: Assuming that we have two 3-mode tensors, namely \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\). Then, the t-product between \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) can be calculated as follows: \[\mathcal{A}_{3}=\mathcal{A}_{1}*\mathcal{A}_{2}=bvfold\{bcirc(\mathcal{A}_{1}) bvec(\mathcal{A}_{2})\}.\] (5)
* **Tensor-Singular Value Decomposition (T-SVD)**: \[\mathcal{A}=\mathcal{Q}_{left}*\mathcal{S}_{f}*\mathcal{Q}_{right}^{\top},\] (6) where \(\mathcal{Q}_{left}\) and \(\mathcal{Q}_{right}\) are the left and right orthonormal singular vectors of the tensor \(\mathcal{A}\), respectively. \(\mathcal{S}_{f}\) is the singular value core tensor.