## 3 Details of the Proposed Algorithm  
### General Framework  
ADMM is an effective method to solve large-scale structured optimization problems in the form  
\[\underset{\mathbf{x},\mathbf{z}}{\text{minimize}}\ f(\mathbf{x})+g(\mathbf{z}) \tag{12}\] \[\text{subject to}\ \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{z}= \mathbf{c}\]  
leveraging the decomposability of dual ascent methods while benefiting from the robust convergence properties of the method of multipliers [25]. Given the objective function in (11), direct optimization is difficult as both \(\ell_{1}\)-norm terms are non-differentiable. By introducing an auxiliary variable \(\mathbf{z}=\mathbf{y}-\mathbf{\Phi}\mathbf{x}\), we can recast (11) into a form amenable to ADMM:  
\[\underset{\mathbf{x},\mathbf{z}}{\text{minimize}}\ \tau||\mathbf{z}||_{1}+|| \mathbf{x}||_{1} \tag{13}\] \[\text{subject to}\ \mathbf{\Phi}\mathbf{x}+\mathbf{z}=\mathbf{y}.\]  
The Augmented Lagrangian Function (ALF) of (13) is  
\[L_{\rho}(\mathbf{x},\mathbf{z},\mathbf{\gamma})=||\mathbf{x}||_{1}+\tau||\mathbf{ z}||_{1}+\frac{\rho}{2}||\mathbf{z}+\mathbf{\Phi}\mathbf{x}-\mathbf{y}+\gamma/\rho||_{2}^ {2}-\frac{1}{2\rho}||\mathbf{\gamma}||_{2}^{2}, \tag{14}\]  
incorporating the dual variable vector \(\mathbf{\gamma}\) and a positive penalty factor \(\rho\). Rather than tackling the minimization of \(L_{\rho}(\mathbf{x},\mathbf{z},\mathbf{\gamma})\) with respect to both \(\mathbf{x}\) and \(\mathbf{z}\) simultaneously, ADMM decomposes the original problem, allowing for the iterative and alternate updating of \(\mathbf{x}\) and \(\mathbf{z}\) by  
\[\mathbf{x}^{(k+1)} =\underset{\mathbf{x}}{\text{argmin}}\ L_{\rho}(\mathbf{x},\mathbf{z}^{(k)},\mathbf{\gamma}^{(k)}) \tag{15a}\] \[\mathbf{z}^{(k+1)} =\underset{\mathbf{z}}{\text{argmin}}\ L_{\rho}(x^{(k+1)},\mathbf{ z},\mathbf{\gamma}^{(k)})\] (15b) \[\mathbf{\gamma}^{(k+1)} =\mathbf{\gamma}^{(k)}+\rho(\mathbf{z}^{(k+1)}+\mathbf{\Phi}\mathbf{x}^{( k+1)}-\mathbf{y}). \tag{15c}\]Central to the iteration process above are the two subproblems shown in (15a) and (15b), which, upon transformation, correspond to  
\[\underset{\mathbf{x}}{\text{minimize }}F(\mathbf{x}) =f_{1}(\mathbf{x})+f_{2}(\mathbf{x}) \tag{16}\] \[\underset{\mathbf{z}}{\text{minimize }}G(\mathbf{z}) =g_{1}(\mathbf{z})+g_{2}(\mathbf{z}) \tag{17}\]  
with  
\[\begin{split} f_{1}(\mathbf{x})&\triangleq\frac{ \rho}{2}||\mathbf{z}^{(k)}+\boldsymbol{\Phi}\mathbf{x}-\mathbf{y}+\gamma^{(k) }/\rho||_{2}^{2},\quad f_{2}(\mathbf{x})\triangleq||\mathbf{x}||_{1},\\ g_{1}(\mathbf{z})&\triangleq\frac{\rho}{2}|| \mathbf{z}+\boldsymbol{\Phi}\mathbf{x}^{(k+1)}-\mathbf{y}+\gamma^{(k)}/\rho|| _{2}^{2},\quad g_{2}(\mathbf{z})\triangleq\tau||\mathbf{z}||_{1}.\end{split} \tag{18}\]  
Both \(F(\mathbf{x})\) and \(G(\mathbf{z})\) comprise a smooth convex quadratic function alongside a non-differentiable \(\ell_{1}\)-norm term, which can be solved by PGM. Given a non-differentiable function \(h(\mathbf{x})\), its proximal mapping is defined as [36]  
\[\text{prox}_{h,t}(\mathbf{x})=\underset{\mathbf{x}^{+}}{\text{argmin }}\frac{1}{2t}||\mathbf{x}^{+}-\mathbf{x}||_{2}^{2}+h(\mathbf{x}^{+}), \tag{19}\]  
where \(t\) is a step-size parameter chosen by the Lipschitz constant of the differentiable portion of the objective function. For the \(\ell_{1}\)-norm regularization function, its proximal operator is equivalent to the soft-thresholding operator  
\[\mathcal{S}_{\alpha}(\beta)\triangleq\frac{\max(|\beta|-\alpha,0)}{\max(| \beta|-\alpha,0)+\alpha}\beta \tag{20}\]  
as the separability of the \(\ell_{1}\)-norm simplifies the proximal mapping problem into a series of one-dimensional minimization tasks [36]. The following subsections will discuss the detailed schemes for solving the two subproblems.