## 3 Proposed Approach for Addressing Problem (5)
### Deep Learning-Based Scheme for Optimizing \(\alpha_{k}\)
The common rate variables specify the number of nodes in the output  
Figure 4: PCA module representation for the channel \(\mathbf{q}_{1,PS}\). layer, \(\{\alpha_{k}\}\). During the training stage, the weights of the DNN are optimized using the backpropagation algorithm [38], given a training dataset with \(\{\alpha_{k}\}\) as the real-valued targets. In conjunction with the grid search method, the K-fold cross-validation method is employed to select the best hyperparameters, including the number of hidden layers, hidden nodes, learning rates, and activation functions. Subsequently, during the online stage, after successfully training the DNN, the predicted common rate variables for the current channel vectors are constrained to feasible ranges, defined by the constraints (5b) and (5g), and are expressed as follows:  
\[\alpha_{k}= \begin{cases}0,&\text{if }\alpha_{k}<0,\ \forall k,\\ \alpha_{k},&\text{if }0\leq\alpha_{k}\leq\chi_{k},\ \forall k,\\ \chi_{k},&\text{otherwise},\ \forall k\end{cases}  \tag{10a}\] \[\alpha_{k}= \begin{cases}0,&\text{if }\alpha_{k}<0,\ \forall k,\\ \alpha_{k},&\text{if }0\leq\alpha_{k}\leq\chi_{k},\ \forall k,\\ \chi_{k},&\text{otherwise},\ \forall k\end{cases}  \tag{10b}\]  
The modified common rate variables are passed to the SDR-based scheme to optimize the PS ratios, \(\theta_{k}\), and the beamforming vectors, \(\mathbf{p}_{0},\left\{\mathbf{p}_{k},\mathbf{e}_{g}\right\}\). Within the DNN, we analyze the computational complexity in the online stage, as the training process takes place offline. This complexity is closely tied to the number of nodes and layers within the DNN. In particular, there are a total of \(L(2K+2G)+2K+G\) nodes in the input layer, \(K\) nodes in the output layer, and each \(i\)th hidden layer has \(N_{H,i}\) nodes, where \(i=1,\ldots,H_{\mathrm{D}}\). Consequently, the computational complexity of the DNN module can be expressed as \(\mathcal{O}((L(2K+2G)+2K+G)N_{H,1}+\ N_{H,1}N_{H,2}+\ldots+N_{H,H_{\mathrm{D}}}K)\), even though this can be approximated to \(\mathcal{O}(L(2K+2G))\) when the DNN parameters are considered fixed. SDR-Based Method for Optimizing \(\mathbf{p}_{0}\), \(\left\{\mathbf{p}_{k},\mathbf{e}_{g},\theta_{k}\right\}\)  
We introduce an approach based on the SDR technique to jointly optimize the PS factors and beamforming vectors, with a fixed value for \(\{\alpha_{k}\}\). The matrix variables are denoted as \(\mathbf{P}_{i}=\mathbf{P}_{i}\mathbf{P}_{i}^{H}\), \(\mathbf{E}_{g}=\mathbf{e}_{g}\mathbf{e}_{g}^{H}\), \(\mathbf{Q}_{k,PS}=\mathbf{q}_{k,PS}\mathbf{q}_{k,PS}^{H}\), \(\mathbf{H}_{k,PS}=\mathbf{h}_{k,PS}\mathbf{h}_{k,PS}^{H}\), \(\mathbf{Q}_{g,EH}=\mathbf{q}_{g,EH}\mathbf{q}_{g,EH}^{H}\), and \(\mathbf{H}_{g,EH}=\mathbf{h}_{g,EH}\mathbf{h}_{g,EH}^{H}\). In addition, the matrix variable \(\mathbf{P}_{i}\) enforces the conditions \(\mathbf{P}_{i}\succeq 0\), \(\mathrm{rank}(\mathbf{P}_{i})=1\), while matrix \(\mathbf{E}_{g}\) enforces \(\mathbf{E}_{g}\succeq 0\) and \(\mathrm{rank}\big{(}\mathbf{E}_{g}\big{)}=1\). Problem (7) can be transformed into a convex problem by removing the rank-one constraints and given \(\{\alpha_{k}\}\), as follows:  
\[\min_{\mathbf{P}_{0},\left\{\mathbf{P}_{i},\theta_{k},\mathbf{E}_ {g}\right\}} \ \sum_{i=0}^{K}\mathrm{Tr}(\mathbf{P}_{i})+\sum_{g=1}^{G}\mathrm{Tr} \big{(}\mathbf{E}_{g}\big{)}\] (11a) subject to \[\left(\sum_{i=1,\ell\
eq k}^{K}\mathrm{Tr}\big{(}\mathbf{Q}_{k,PS} \mathbf{P}_{i}\big{)}+\sum_{g=1}^{G}\mathrm{Tr}\big{(}\mathbf{H}_{k,PS} \mathbf{E}_{g}\big{)}+\sigma_{k}^{2}+\frac{\gamma_{k}^{2}}{\theta_{k}}\right) \omega_{k}-\mathrm{Tr}(\mathbf{Q}_{k,PS}\mathbf{P}_{k})\leq 0,\,\forall k \tag{11b}\]  
Figure 5: DNN module scheme for the proposed approach.