# AI-Driven Pilot Overhead Reduction in 5G mmWaveMassive MIMO Systems
## 2 System Model
### AI-Driven Pilot Placement
#### 2.2.2 Supervised ML
By adapting the equations to the context of a multi-user MIMO system with \(M\) transmitter antennas, \(N\) receiver antennas, and \(K\)data streams, the following equation can be obtained:  
\[G=\sum_{i=1}^{K}p(i)(1-p(i)) \tag{14}\]  
where \(G\) is the Gini impurity factor, which is a measure of how often a randomly chosen element from the set would be incorrectly labeled if it were randomly labeled according to the distribution of labels in the subset, and \(p(i)\) represents the probability of user \(i\) at a node in the decision tree. The entropy, which is a measure of uncertainty or disorder in a system equation, is given by the following:  
\[H=\sum_{i=1}^{K}p(i)log_{2}(p(i)) \tag{15}\]  
The mean square error \(MSE\) is also a useful parameter in random forest regression used as a loss function for assessing the accuracy of predictions made by the model given by the following equation:  
\[MSE=\frac{1}{n}{\sum_{i=1}^{n}}{\left(y(i)-\hat{y}(i)\right)^{2}} \tag{16}\]  
where \(n\) represents the number of samples in the training data, \(y(i)\) represents the actual pilot overhead, and \(\hat{y}(i)\) represents the predicted pilot overhead. The final prediction is given by the equation below:  
\[\hat{y}=\frac{1}{N}{\sum_{i=1}^{N}}\hat{y}(i) \tag{17}\]  
In the equations provided, \(M\) (the number of transmitter antennas) does not directly appear in the decision tree splitting criterion or the aggregation of predictions. However, \(M\) indirectly affects the feature space used in training the random forest regression model. The features used in training the model include characteristics related to the transmitter antennas, such as antennas' spatial distribution and transmit power levels. These features can indirectly influence the decision tree splitting process and the model's predictions, which are noted in the Results Section of this study. Neural Networks SVD Enhancement  
Neural networks (NNs) are a cornerstone of machine learning, designed to mimic the human brain's structure and function, where these networks consist of layers of interconnected nodes, giving an analogy of neurons, which process input data through a series of connections and transformations. They can learn and model complex relationships between inputs and outputs through training on datasets. The process of learning takes place through the adjustment of connection weights using back-propagation in order to reduce errors, wherein the neural network acquires knowledge from the training data by minimizing the difference between its predicted outcome and the real outcome [21; 22]. Integrating SVD hybrid beamforming with NN regression can involve using NNs to predict the optimal beamforming vectors or other relevant parameters based on the channel's characteristics. This prediction is based on historical data, where the NN learns the relationship between the channel conditions (input) and the optimal SVD-based beamforming strategies (output). The input to the neural network can include features derived from the channel matrix \(H\), which are singular values and angles of arrival in the proposed model. Let \(X\) represent the input matrix to the NN, with dimensions \(K\times P\) and with \(P\) representing the number of input features. The neural network predicts the optimal beamforming parameters or vectors. \(Y\) denotes the output matrix from the neural network, with dimensions \(K\times Q\), where \(Q\) represents the number of output parameters or dimensions of the beamforming vectors. The training dataset consists of pairs of input-output examples, where the input is derived from the channel conditions, and the output is the corresponding optimal beamforming parameters obtained through SVD-based hybrid beamforming. During training, the neural network minimizes a loss function, in our case the MSE, to optimize the predictions relative to the ground truth obtained from SVD-based hybrid beamforming.