## 3 Proposed Approach for Addressing Problem (5)
### Deep Learning-Based Scheme for Optimizing \(\alpha_{k}\)  
This subsection describes the PCA module for dimensionality reduction and the DNN-based method to predict the common rate variables, \(\alpha_{k}\). Figure 4 presents the PCA module used to derive a reduced representation of each channel vector. The PCA technique [36] projects the input data onto a lower-dimensional subspace to maximize the variance of the projected data. The training dataset is defined  
Figure 3: Overall procedure of the proposed approach. as \(\mathbf{X}=\left\{\mathbf{x}_{1},\ldots,\mathbf{x}_{n},\ldots,\mathbf{x}_{N_{D}}\right\}\), where \(N_{D}\) is the total number of channel vectors used as the training dataset, and \(\mathbf{x}_{n}\) is a \(W\)-dimensional vector composed of the real and imaginary components of the \(n\)th channel. The PCA module maps a channel vector onto an \(L\)-dimensional subspace that satisfies \(L<W\). The projection of \(\mathbf{x}_{n}\) can be represented by \(\mathbf{y}_{n}\)\(=\)\(\mathbf{U}_{L}^{T}\mathbf{x}_{n}\), where \(\mathbf{U}_{L}=\left[\mathbf{u}_{1}^{T},\ldots,\mathbf{u}_{L}^{T}\right]\) with \(\mathbf{u}_{1}^{T}\mathbf{u}_{l}=1\) for \(l=1,\ldots,L\). A \(\mathbf{U}_{L}\) that maximizes the trace of the covariance matrix of \(\{\mathbf{y}_{n}\}\) is needed to maximize the variance of \(\{\mathbf{y}_{n}\}\), which can be expressed as:  
\[\mathbf{U}_{L}^{*}=\underset{\mathbf{U}_{L}}{\arg\max}\big{(}\text{tr}\big{(} \mathbf{C_{y}}\big{)}\big{)}, \tag{8}\]  
where \(\mathbf{C_{y}}=\frac{1}{N_{D}}\sum\limits_{n=1}^{N_{D}}\left(\mathbf{y}_{n}- \bar{\mathbf{y}}\right)\)\(\left(\mathbf{y}_{n}-\bar{\mathbf{y}}\right)^{T}\),and \(\bar{\mathbf{y}}=\frac{1}{N_{D}}\sum\limits_{n=1}^{N_{D}}\mathbf{x}_{n}\). Let \(\mathbf{C_{x}}=\frac{1}{N_{D}}\sum\limits_{n=1}^{N_{D}}\left(\mathbf{x}_{n}- \bar{\mathbf{x}}\right)\)\(\left(\mathbf{x}_{n}-\bar{\mathbf{x}}\right)^{T}\) represent the covariance matrix of \(\{\mathbf{x}_{n}\}\), and \(\bar{\mathbf{x}}=\frac{1}{N_{D}}\sum\limits_{n=1}^{N_{D}}\mathbf{x}_{n}\). The solution of (8), described in [36; 37], is obtained when  
\[\mathbf{C_{x}}\mathbf{u}_{l}=\lambda_{l}\mathbf{u}_{l}, \tag{9}\]  
which means that the variance of the projected data is maximized when \(\mathbf{u}_{l}\) is an eigenvector of \(\mathbf{C_{x}}\). Therefore, the optimal \(\mathbf{U}_{L}^{*}\) is a matrix composed of the first \(L\) eigenvectors of \(\mathbf{C_{x}}\) as columns. Algorithm 1 summarizes the steps for performing the PCA technique. ```
1:inputs: Training dataset \(\mathbf{X}=\left\{\mathbf{x}_{1},\ldots,\mathbf{x}_{n},\ldots,\mathbf{x}_{N_{D}}\right\}\) containing the wireless channels and the number of components to keep, \(L\). 2: Evaluate the mean \(\bar{\mathbf{x}}\). 3: Calculate the covariance matrix \(\mathbf{C_{x}}\). 4: Decompose \(\mathbf{C_{x}}\) to obtain the eigenvectors and eigenvalues. 5: Select \(L\) eigenvectors corresponding to the \(L\) largest eigenvalues and create \(\mathbf{U}_{L}\). 6: Map the \(W\)-dimensional channel vector into an \(L\)-dimensional representation by \(\mathbf{y}_{n}=\mathbf{U}_{L}^{T}\mathbf{x}_{n}\). 7:output: Reduced representation of the channel vectors \(\mathbf{q}_{k,PS},\mathbf{q}_{g,EH},\mathbf{h}_{k,PS},\mathbf{h}_{g,EH}\) by performing step 6 for each channel vector. ```  
**Algorithm 1** PCA for dimensionality reduction  
Figure 5 shows the DNN model for the proposed approach, composed of an input layer, \(H_{D}\) hidden layers, and an output layer. The number of nodes in the input layer is determined by the user's rate requirements, EH requirements, and the reduced representation of the channel vectors after the PCA module, \(\left\{\mathbf{PC}_{\mathbf{q}_{k,PS}},\mathbf{PC}_{\mathbf{q}_{g,EH}}, \mathbf{PC}_{\mathbf{h}_{k,PS}},\mathbf{PC}_{\mathbf{h}_{g,EH}}\right\}\). Each hidden layer consists of \(N_{H}\) hidden nodes, where the number of hidden layers and the number of hidden nodes are determined by fine-tuning the hyperparameters, as discussed in Section 4.