## 3 The Method Proposed in This Paper
### ADMM-1 DNet
 \tag{18}\]  
where  
\[W=\rho\mathbf{\Phi}\Big{(}\mathbf{A}^{T}\mathbf{A}+\rho\mathbf{\Phi}^{T}\mathbf{\Phi}\Big{)}^{-1} \mathbf{\Phi}^{T}\in\mathbb{R}^{N\times N}, \tag{19}\]  
\[b=b(y)=\mathbf{\Phi}\big{(}\mathbf{A}^{T}\mathbf{A}+\rho\mathbf{\Phi}^{T}\mathbf{\Phi}\big{)}^{-1} \mathbf{A}^{T}y\in\mathbb{R}^{N\times 1}. \tag{20}\]  
We introduce \(v^{(k)}=\left[u^{(k)};\,z^{(k)}\right]\in\mathbb{R}^{2N\times 1}\) and set \(\Theta=(-I-W|W)\in\mathbb{R}^{2N\times 1}\) and \(\Lambda=\big{(}I-W|W)\in\mathbb{R}^{N\times 2N}\) to get the following:  
\[v_{(k+1)}= \begin{pmatrix}\Lambda\\ \mathcal{O}_{N\times 2N}\end{pmatrix} v_{(k)}+ \begin{pmatrix}b\\ 0\end{pmatrix} + \begin{pmatrix}-S_{\lambda/\rho}(\Theta v_{(k)}+b)\\ S_{\lambda/\rho}(\Theta v_{(k)}+b)\end{pmatrix} . \tag{21}\]  
Setting \(\widetilde{\Theta}=[\Lambda;\,\mathcal{O}_{N\times 2N}]\in\mathbb{R}^{2N \times 2N}\), \(I_{1}=[I_{N\times N};\,\mathcal{O}_{N\times N}]\in\mathbb{R}^{2N\times 2N}\) and \(I_{2}=[-I_{N\times N};\,I_{N\times N}]\in\mathbb{R}^{2N\times N}\), transform Equation (21) into the following:  
\[v_{(k+1)}=\widetilde{\Theta}v_{(k)}+I_{1}b+I_{2}S_{\lambda/\rho}(\Theta v_{(k) }+b). \tag{22}\]  
Based on the above equation, the ADMM algorithm is expanded to an \(L\)-layer neural network is defined as follows:  
\[f_{1}(y)=I_{1}b(y)+I_{2}S_{\lambda/\rho}(b(y)), \tag{23}\]  
\[f_{k}(v)=\widetilde{\Theta}v+I_{1}b+I_{2}S_{\lambda/\rho}(\Theta v+b),k=2, \ldots,L \tag{24}\]  
where \(\mathbf{\Phi}\) is trained as a trainable parameter, denote \(L\) such layers (all having the same \(\mathbf{\Phi}\)) as  
\[f_{\Phi}^{L}(y)=f_{L}\circ\cdots\circ f_{1}(y). \tag{25}\]  
Applying the radiation mapping \(T\) driven by Equation (15) to the final layer of the neural network yields the final output \(\hat{x}\), i.e.,  
\[\hat{x}=T\Big{(}f_{\Phi}^{L}(y)\Big{)}=\Big{(}\mathbf{A}^{T}\mathbf{A}+\rho\mathbf{\Phi}^ {T}\mathbf{\Phi}\Big{)}^{-1}\Big{(}\mathbf{A}^{T}y+\rho\mathbf{\Phi}^{T}\Big{(}z^{L}-u^{L }\Big{)}\Big{)}, \tag{26}\]  
where \(\left[u^{L};\,z^{L}\right]=v_{L}\), in order to clip the output when the number of paradigms is out of range, it is necessary to add an additional function \(\sigma:\,\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) defined as: if \(\left\|x\right\|_{2}\leq B_{\text{out}}\), then \(\sigma(x)=x\), otherwise \(\sigma(x)=B_{\text{out}}x/\left\|x\right\|_{2}\), where \(B_{\text{out}}>0\) is a fixed constant. A hypothetical class is introduced containing the functions used that can be realized by ADMM-1 DNet:  
\[H^{L}=\Big{\{}\sigma\circ h:\mathbb{R}^{m}\mapsto\mathbb{R}^{n}:h(y)=T(f_{ \Phi}^{L}(y)),\,\,\,\mathbf{\Phi}\in\mathbb{R}^{N\times n},N>n\Big{\}}. \tag{27}\]  
Given the above-hypothesized class and a set \(S=\big{\{}(y_{i},x_{i})\big{\}}_{i=1}^{s}\) containing \(S\) training samples, ADMM-1 DNet yields a function \(h_{s}\in H^{L}\) that aims at reconstructing \(x\) from \(y=Ax\). Figure 2 gives a schematic diagram of ADMM-1 DNet, where the operations of each block follow Equations (23) and (24). Input \(\mathbf{\Phi}\), \(\rho\), and \(\lambda\) into the ADMM-1 DNet network as parameters. So that they can learn adaptively rather than remaining constant during the training process. So here \(\mathbf{\Phi}^{T}\) is actually not a transpose of \(\mathbf{\Phi}\) in the strict sense.