## 2 Background
### Optimization of the Proposed Objective Function
The solution of Equation (15) can be found by applying tensor tubal-shrinkage operator [26]: \[\mathcal{H}_{k+1}=\mathcal{F}_{\tau}\big{(}\mathcal{P}_{k}\big{)}=\mathcal{Q} _{left}*\mathcal{F}_{\tau}\big{(}\mathcal{S}_{f}\big{)}*\mathcal{Q}_{right}^{ \top},\] (16) where \(\mathcal{P}_{k}=\mathcal{Q}_{left}*\mathcal{S}_{f}*\mathcal{Q}_{right}^{\top}\), \(\mathcal{F}_{\tau}\big{(}\mathcal{S}_{f}\big{)}=\mathcal{S}_{f}*\mathcal{D}\) and \(\tau=\frac{M}{\beta_{2}}\). The diagonal tensor \(\mathcal{D}\in\mathbb{R}^{n\times n\times M}\) contains the singular values in the frequency domain and can be calculated as \(\mathcal{D}(i,i,j)=(1-\frac{\tau}{\mathcal{S}_{f}^{(i)}(i,i)})_{+}\). * **Updating E**: The sub-problem of E can be written as: \[ \begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{m=1}^{M}&\bigg{(}\langle\mathbf{X}_{1_{k}}^{(m)},\mathbf{A}^{(m)}-\mathbf{A}^{(m)}\mathbf{Z}_{k}^{(m)}-\mathbf{E}_{k}^{(m)} \rangle\\ &\quad+\frac{\beta_{1}}{2}\|\mathbf{A}^{(m)}-\mathbf{A}^{(m)} \mathbf{Z}_{k}^{(m)}-\mathbf{E}_{k}^{(m)}\|_{F}^{2}\bigg{)},\end{split} \] (17) where Equation (17) can be written as: \[\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E}\|_{2,1}+\sum_{m=1}^{M} \frac{\beta_{1}}{2}\|\mathbf{E}^{(m)}-\mathbf{N}_{k}^{(m)}\|_{F}^{2},\] (18) where \(\mathbf{N}_{k}^{(m)}=\mathbf{A}^{(m)}-\mathbf{A}^{(m)}\mathbf{Z}_{k}^{(m)}+ \frac{\mathbf{X}_{1_{k}}^{(m)}}{\beta_{1}}\). To simplify Equation (18), we construct a matrix \(\mathbf{N}\) by stacking the layers of \(\mathbf{N}^{(m)},m=\{1,\dots,M\}\) vertically which leads to the following equation: \[\mathbf{E}_{k+1}=\min_{\mathbf{E}}\quad\gamma\|\mathbf{E}\|_{2,1}+\frac{\beta _{1}}{2}\|\mathbf{E}-\mathbf{N}\|_{F}^{2},\] (19) where E can be computed as suggested in [27]: \[\mathbf{E}_{k+1}=\left\{ \begin{array}{cc}\frac{\|\mathbf{N}_{i,j}\|_{2}- \frac{\gamma}{\beta_{1}}}{\|\mathbf{N}_{i,j}\|_{2}}&\|\mathbf{N}_{.j}\|_{2}> \frac{\gamma}{\beta_{1}}\\ 0&\text{elsewhere}\end{array} \right.,\] (20) where \(\|\mathbf{N}_{. ;j}\|_{2}\) refers to the \(L_{2}\)-norm of the \(j^{th}\) column in \(\mathbf{N}\). * **Updating \(\mathbf{Z}^{(m)}\)**: Considering all the terms with \(\mathbf{Z}^{(m)}\) and fixing the rest of the variables, the sub-problem of \(\mathbf{Z}_{k+1}^{(m)}\) is written as: \[ \begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{m=1}^{M}&\bigg{(}\langle\mathbf{X}_{1_{k}}^{(m)},\mathbf{A}^{( m)}-\mathbf{A}^{(m)}\mathbf{Z}_{k}^{(m)}-\mathbf{E}_{k}^{(m)}\rangle\\ &\quad+\frac{\beta_{1}}{2}\|\mathbf{A}^{(m)}-\mathbf{A}^{(m)}\mathbf{Z}_{ k}^{(m)}-\mathbf{E}_{k}^{(m)}\|_{F}^{2}\bigg{)},\end{split} \] (21) where Equation (17) can be written as: \[ \begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{m=1}^{M}&\frac{\beta_{1}}{2}\|\mathbf{E}^{(m)}- \mathbf{N}_{k}^{(m)}\|_{F}^{2},\end{split} \] (22) where \(\mathbf{N}_{k}^{(m)}=\mathbf{A}^{(m)}-\mathbf{A}^{(m)}\mathbf{Z}_{k}^{(m)}+ \frac{\mathbf{X}_{1_{k}}^{(m)}}{\beta_{1}}\). To simplify Equation (18), we construct a matrix \(\mathbf{N}\) by stacking the layers of \(\mathbf{N}^{(m)},m=\{1,\dots,M\}\) vertically which leads to the following equation: \[\mathbf{E}_{k+1}=\min_{\mathbf{E}}\quad\gamma\|\mathbf{E}\|_{2,1}+\frac{\beta _{1}}{2}\|\mathbf{E}-\mathbf{N}\|_{F}^{2},\] (19) where E can be computed as suggested in [27]: \[\mathbf{E}_{k+1}=\left\{ \begin{array}{cc}\frac{\|\mathbf{N}_{i,j}\|_{2}- \frac{\gamma}{\beta_{1}}}{\|\mathbf{N}_{i,j}\|_{2}}&\|\mathbf{N}_{.j}\|_{2}> \frac{\gamma}{\beta_{1}}\\ 0&\text{elsewhere}\end{array} \right.,\] (23) where \(\|\mathbf{N}_{. ;j}\|_{2}\) refers to the \(L_{2}\)-norm of the \(j^{th}\) column in \(\mathbf{N}\). * **Updating \(\mathbf{Z}^{(m)}\)**: Considering all the terms with \(\mathbf{Z}^{(m)}\) and fixing the rest of the variables, the sub-problem of \(\mathbf{Z}_{k+1}^{(m)}\) is written as: \[