## 4 Proposed Methods
### Update of \(\mathbf{X}\)  
Given fixed \(\rho\) and \(\mathbf{X}_{0}\), the optimization problem Equation (11) for \(\mathbf{X}\) is defined as follows  
\[ \begin{split}&\min_{\mathbf{X}}\,(1-\delta)\|\mathbf{S}-\rho \mathbf{H}\mathbf{X}\|_{F}^{2}+\delta\|\mathbf{X}-\mathbf{X}_{0}\|_{F}^{2}\\ &\text{s.t. }x_{b,l}\in\mathcal{X},\;\forall b,l\end{split}  \tag{17}\]  
Let \(\mathbf{C}=\left[\sqrt{1-\delta}\rho\mathbf{H}^{T},\sqrt{\delta}\mathbf{I}_{ B}\right]^{T}\in\mathbb{C}^{(U+B)\times B}\) and \(\mathbf{D}=\left[\sqrt{1-\delta}\mathbf{S}^{T},\sqrt{\delta}\mathbf{X}_{0} ^{T}\right]^{T}\in\mathbb{C}^{(U+B)\times L}\); then, problem Equation (17) can be represented more compactly as  
\[ \begin{split}&\min_{\mathbf{X}}\|\mathbf{C}\mathbf{X}-\mathbf{D} \|_{F}^{2}\\ &\text{s.t. }x_{b,l}\in\mathcal{X},\;\forall b,l\end{split}  \tag{18}\]  
Let \(\mathbf{x}=\text{vec}(\mathbf{X})\in\mathbb{C}^{LB\times 1}\) and \(\mathbf{d}=\text{vec}(\mathbf{D})\in\mathbb{C}^{L(U+B)\times 1}\); thus, the equivalent expression of problem Equation (18) can be transformed into  
\[ \begin{split}&\min_{\mathbf{x}}\left\|\bar{\mathbf{C}}\bar{ \mathbf{x}}-\bar{\mathbf{d}}\right\|_{2}^{2}\\ &\text{s.t. }x_{b,l}\in\mathcal{X},\;\forall b,l\end{split}  \tag{19}\]  
where \(\bar{\mathbf{C}}= \begin{bmatrix}\Re(\sqrt{\rho_{\text{T}}/2B}\mathbf{I}_{L} \otimes\mathbf{C})&-\Im(\sqrt{\rho_{\text{T}}/2B}\mathbf{I}_{L} \otimes\mathbf{C})\\ \Im(\sqrt{\rho_{\text{T}}/2B}\mathbf{I}_{L}\otimes\mathbf{C})&\Re(\sqrt{\rho_{\text{T}}/2B} \mathbf{I}_{L}\otimes\mathbf{C})\end{bmatrix} \), \(\bar{\mathbf{x}}= \begin{bmatrix}\Re(\mathbf{x})&\Im(\mathbf{x})\end{bmatrix} ^{T}\), and \(\bar{\mathbf{d}}= \begin{bmatrix}\Re(\mathbf{d})&\Im(\mathbf{d})\end{bmatrix} ^{T}\). By utilizing the ADMM framework, problem Equation (19) can be rewritten into an equivalent form as  
\[ \begin{split}&\min_{\mathbf{z},\mathbf{x}}\left\|\bar{\mathbf{C}} \bar{\mathbf{z}}-\bar{\mathbf{d}}\right\|_{2}^{2}+I_{\chi}(\bar{\mathbf{x}}) \\ &\text{s.t. }\bar{\mathbf{z}}-\bar{\mathbf{x}}=0\end{split}  \tag{20}\]  
where \(\bar{\mathbf{z}}\) denotes the auxiliary vector that must satisfy the corresponding constraints. The function \(I_{\chi}(\bar{\mathbf{x}})\) is the indicator function for the set \(\mathcal{X}\), where \(I_{\chi}(\bar{\mathbf{x}})\) equals 0 when \(\bar{\mathbf{x}}\) is in \(\mathcal{X}\) and \(\infty\) otherwise. Utilizing the augmented Lagrangian multiplier method, we can reformulate Equation (20) as follows  
\[L(\bar{\mathbf{z}},\bar{\mathbf{x}})=\left\|\bar{\mathbf{C}}\bar{\mathbf{z}}- \bar{\mathbf{d}}\right\|_{2}^{2}+I_{\chi}(\bar{\mathbf{x}})+2\eta^{H}(\bar{ \mathbf{z}}-\bar{\mathbf{x}})+c\|\bar{\mathbf{z}}-\bar{\mathbf{x}}\|_{2}^{2} \tag{21}\]where the term \(c\) denotes the penalty parameter and \(\mathbf{\eta}\) denotes the dual vector. We employ the ADMM to solve Equation (20) through the following three steps [38]  
\[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\] \[\bar{\mathbf{x}}^{(k+1)} =\arg\min_{\bar{\mathbf{x}}}L(\bar{\mathbf{z}}^{(k+1)},\bar{\mathbf{x}},\mathbf{ \eta}^{(k)})\] (22b) \[\mathbf{\eta}^{(k+1)} =\mathbf{\eta}^{(k)}+c(\bar{\mathbf{z}}^{(k+1)}-\bar{\mathbf{x}}^{(k+1)}) \tag{22c}\]  
After performing some algebraic manipulations, Equation (22) can be explicitly written as  
\[\bar{\mathbf{z}}^{k+1} =\left(\bar{\mathbf{C}}^{H}\bar{\mathbf{C}}+c\mathbf{\mathrm{I}}_{2 LB}\right) ^{-1}(c\mathbf{x}^{(k)}+\bar{\mathbf{C}}^{H}\bar{\mathbf{d}}-\mathbf{\eta}^{(k)}) \tag{23a}\] \[\bar{\mathbf{x}}^{(k+1)} =g_{\mathcal{X}}(\bar{\mathbf{z}}^{(k+1)}+\frac{1}{c}\mathbf{\eta}^{(k)})\] (23b) \[\mathbf{\eta}^{(k+1)} =\mathbf{\eta}^{(k)}+c(\bar{\mathbf{z}}^{(k+1)}-\bar{\mathbf{x}}^{(k+1)}) \tag{23c}\]  
where the term \(g_{\mathcal{X}}(\cdot)\) represents the projection onto \(x_{b,l}\in\mathcal{X},~{}\forall b,l\).