## 2 Background
### Optimization of the Proposed Objective Function
\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} \] (24) where Equation (17) can be written as: \[ \begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{m=1}^{M}&\frac{\beta_{1}}{2}\|\mathbf{E}^{(m)}-\mathbf{N} _{k}^{(m)}\|_{F}^{2},\end{split} \] (25) where Equation (18) is rewritten as: \[ \begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{m=1}^{M}&\frac{\beta_{1}}{2}\|\mathbf{E}^{(m)}-\mathbf{N} _{k}^{(m)}\|_{F}^{2},\end{split} \] (26) where Equation (19) is rewritten as: \[ \begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{m=1}^{M}&\frac{\beta_{1}}{2}\|\mathbf{E}^{(m)}- \mathbf{N}_{k}^{(m)}\|_{F}^{2},\end{split} \] (27) where Equation (28) is rewritten as: \[ \begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{m=1}^{M}&\frac{\beta_{1}}{2}\|\mathbf{E}^{(m)}- \mathbf{N}_{k}^{(m)}\|_{F}^{2},\end{split} \] (28) where Equation (29) is rewritten as: \[\begin{split}\mathbf{E}_{k+1}=\min_{\mathbf{E}}\gamma\|\mathbf{E} \|_{2,1}+\sum_{\[ \begin{split}&\mathbf{Z}_{k+1}^{(m)}=\min_{\mathbf{Z}^{(m)}}\,\langle \mathbf{X}_{1k}^{(m)},\mathbf{A}^{(m)}-\mathbf{A}^{(m)}\mathbf{Z}_{k}^{(m)}- \mathbf{E}_{k+1}^{(m)}\rangle+\frac{\beta_{1}}{2}\|\mathbf{A}^{(m)}-\mathbf{A }^{(m)}\mathbf{Z}_{k}^{(m)}-\mathbf{E}_{k+1}^{(m)}\|_{F}^{2}\\ &+\langle\mathcal{X}_{2},\mathcal{Z}_{k}-\mathcal{H}_{k+1} \rangle+\frac{\beta_{2}}{2}\|\mathcal{Z}_{k}-\mathcal{H}_{k+1}\|_{F}^{2}- \lambda\;tr\Bigg{(}\mathbf{U}_{k}^{(m)}{}^{\top}\mathbf{Z}_{N}{}_{k}^{(m)} \mathbf{U}_{k}^{(m)}\Bigg{)}.\end{split}  \tag{21}\]  
The updated \(\mathbf{Z}^{(m)}\) is calculated by finding the gradient of Equation (21) and setting it to zero [28] which results in a closed form solution of \(\mathbf{Z}_{l+1}^{(m)}\) as:  
\[ \begin{split}&\mathbf{Z}_{k+1}^{(m)}=\,\left(\beta_{2}\mathbf{I}+ \beta_{1}\mathbf{A}^{(m)}{}^{\top}\mathbf{A}^{(m)}\right)^{-1}\cdot\left( \lambda\mathbf{D}^{(m)}{}^{(-0.5)}\mathbf{U}_{k}^{(m)}{}^{\top}\mathbf{D}^{(m )}{}^{(-0.5)}+\beta_{2}\mathbf{H}_{k+1}^{(m)}\right.\\ &-\mathbf{X}_{2k}^{(m)}+\beta_{1}\mathbf{A}^{(m)}\mathbf{T}_{k}^ {(m)}\Bigg{)},\end{split}  \tag{22}\]  
where \(\mathbf{T}_{k}^{(m)}=\mathbf{A}^{(m)}-\mathbf{E}_{k+1}^{(m)}+\frac{\mathbf{x }_{1k}^{(m)}}{\beta_{1}}\) and the matrix \(\mathbf{Z}^{(m)}\) is found for each layer separately. * **Updating \(\mathbf{U}^{(m)}\)**: Fixing all the variables except \(\mathbf{U}^{(m)}\) in Equation (14), we obtain: \[ \begin{split}&\mathbf{U}_{k+1}^{(m)}=\min_{\mathbf{U}^{(m)}}\, -\lambda tr\Bigg{(}\mathbf{U}^{(m)}{}^{\top}\mathbf{Z}_{N}^{(m)}\mathbf{U}^{( m)}\Bigg{)}+w_{m}tr\Bigg{(}\mathbf{U}^{*}{}^{\top}\mathbf{U}^{(m)}{}^{\top} \mathbf{U}^{(m)}{}^{\top}\mathbf{U}^{*}\Bigg{)}\\ &\text{s.t.}\quad\mathbf{U}^{(m)}{}^{\top}\mathbf{U}^{(m)}= \mathbf{I}.\end{split} \] (23) An equivalent form of Equation (23) is: \[ \begin{split}&\mathbf{U}_{k+1}^{(m)}=\max_{\mathbf{U}^{(m)}}\, \lambda\;tr\Bigg{(}\mathbf{U}^{(m)}{}^{\top}\mathbf{Z}_{mod}^{(m)}\mathbf{U} ^{(m)}\Bigg{)}\\ &\text{s.t.}\quad\mathbf{U}^{(m)}{}^{\top}\mathbf{U}^{(m)}= \mathbf{I},\end{split} \] (24) where \(\mathbf{Z}_{mod}^{(m)}=\mathbf{D}^{(m)}{}^{(-0.5)}\mathbf{Z}^{(m)}\mathbf{D} ^{(m)}{}^{(-0.5)}+w_{m}\mathbf{U}^{*}{}^{\top}\). The resultant problem in Equation (24) is equivalent to the maximization problem in Equation (9), but with an adjusted or modified normalized adjacency matrix. Consequently, by the [PERSON] theorem, Equation (24) can be solved through eigenvalue decomposition (EVD) of \(\mathbf{Z}_{mod}^{(m)}\), where \(\mathbf{U}^{(m)}\) contains the set of orthonormal eigenvectors that correspond to the largest \(c\) eigenvalues. * **Updating \(\mathbf{U}^{*}\)**: The \(\mathbf{U}^{*}\) sub-problem can be defined as: \[