# Support Vector Machine-Based Transmit Antenna Allocation for Multiuser Communication Systems
## 3 SVM-Based Antenna Allocation
### Task 1: Designing a Training Sample Set
#### 3.1.2 Subtask 1-2. Training Set Design  
The training samples are the input for a learning system and are known as input variables, predictors, or attributes. As shown in Section 2.1, we assume that magnitude channel information, i.e., the channel gain matrix \(\mathbf{G}\), is available in our communication system. Based on the available channel information, it is important to properly design the training set by taking into account not only the target KPI, which affects communication performance, but also the complexity of the system. For example (refer to [54] and the references therein), singular values are used for a singular value-based antenna selection system, the minimum eigenvalues of the Hermitian matrix of the channel matrix for Gerschgorin circle-based antenna selection systems, channel norm values for norm-based antenna  
Figure 2: Machine learning framework for antenna allocation in a multiuser communication system. selection systems, and the dot products of channel column vectors for correlation-based antenna selection systems. Here, the singular values are clearly a good candidate for a training set for various KPIs (e.g., spectral efficiency, energy efficiency, and BER), yet they require a higher complexity higher than the other values. In this study, we adopt a signal-to-interference-leakage ratio (SILR) metric that is closely coupled with the sum rate, which is our target KPI. It is worth noting that with this SILR metric, it is sufficient to perform antenna allocation with the channel gain knowledge at the Tx, even without knowing the signal-to-interference-plus-noise ratio (SINR) of the users. We also note that it may not be possible to acquire the received SINR at the Tx under our communication mechanism using the channel gain feedback. This is because the sum of interference links can be computed after the antenna allocation process and the set of allocated antennas is not available in the training phase. An SILR matrix, \(\mathbf{Z}\in\mathbb{R}^{U\times N_{t}}\), is employed to generate the training set, which can be computed based on \(\mathbf{G}\). Specifically, the entry of the \(i\)th row and the \(j\)th column of \(\mathbf{Z}\) is given by  
\[Z_{i,j}=\frac{g_{i,j}}{\sum_{k\
eq j}^{U}g_{i,k}}. \tag{5}\]  
From Equation (5), it can be seen that the SILR metric simultaneously captures both the desired signal strength and the interference leakage to other users. Because a machine learning system requires the real-value vector input of multiple features, we transform the SILR matrix into a \(\mathbb{R}^{1\times N_{t}U}\) vector by stacking \(U\) users' vectors (the row vectors in \(\mathbf{Z}\)). The training set vector with \(N_{t}U\) features is given by  
\[[Z_{1,1},\cdots,Z_{1,N_{t}},\cdots,\underbrace{Z_{i,1},\cdots,Z_{i,N_{t}}}_{ \text{the $i$th row of $\mathbf{Z}$}},\cdots,Z_{U,1},\cdots,Z_{U,N_{t}}]. \tag{6}\]  
By repeating the channel generation and data processing \(D\) times, we obtain a training set matrix \(\mathbf{T}_{\text{raw}}\in\mathbb{R}^{D\times N_{t}U}\) whose rows are given by the training vector in Equation (6). As a special case, when \(U=1\) (single-user communication systems), we use the channel gain as the metric (i.e., \(Z_{i,j}=g_{i,j}\)) due to the absence of interferences. Since \(\mathbf{Z}\) is a vector for \(U=1\), we can skip the aforementioned stacking step by directly using \(\mathbf{Z}\) as the training set vector. The final step of training set design is to normalize the training samples to obtain a proper input training set for the learning system. Let \(\mathbf{T}\in\mathbb{R}^{D\times N_{t}U}\) denote a training set matrix as one of the inputs to the learning system, whose \((i,j)\)th element, denoted by \(T_{i,j}\), is a normalized value of the \((i,j)\)th element of \(\mathbf{T}_{\text{raw}}\).