# Support Vector Machine-Based Transmit Antenna Allocation for Multiuser Communication Systems
## 4 Numerical Evaluation
### Sum-Rate Performance  
The sum-rate performance of SVM designed in the previous section is compared to the other schemes. In Figure 5, we illustrate the cumulative density function (CDF) of the sum rate when \(N_{t}=5\) and \(U=3\). It can be seen that the SVM classifier provides the closest sum-rate performance to that of OPT. The \(k\)-NN classifier achieves superior sum-rate performance to that of RAND, yet it is highly inferior to the proposed SVM.  
In Figures 6 and 7, we illustrate the sum rate over SNR for different numbers of users \(U\in\{1,3\}\), when \(N_{t}=5\). For \(U=1\) corresponding to single-user communications with no interference, the sum rate curves of SVM and \(k\)-NN almost coincide with that of OPT due to the excellent classification accuracy of our learning system. On the other hand, for \(U=3\), we observe that SVM outperforms \(k\)-NN, as shown in Figure 5. The sum-rate performance of all methods gets saturated in a high SNR regime because our communication system is interference-limited.  
Figure 4: SVM performance evaluation results when \(N_{t}=5\) and \(U=3\): runtime over \(L^{\prime}\) for various SVM kernel functions.  
In Figure 8, we plot the sum rate over \(N_{t}\) from 5 to 10 when \(U=3\). The number of classes \(L\) is set according to the value of \(N_{t}\), e.g., \(L=10!/7!=720\) for \(N_{t}=10\). We observe that the sum rate of SVM and OPT increases with \(N_{t}\) with the help of the multi-antenna selection diversity gain. Interestingly,  
Figure 5: Empirical CDF of sum rate when \(N_{t}=5\) and \(U=3\).  
Figure 6: Sum rate over signal-to-noise ratio (SNR) performance for different system configurations when \(N_{t}=5\) and \(U=1\).  
Figure 7: Sum rate over SNR performance for different system configurations when \(N_{t}=5\) and \(U=3\).  
the sum-rate performance of \(k\)-NN tends to saturate with increasing \(N_{t}\) due to the limited number of training samples. More specifically, while there are 720 classes, the number of training samples is not sufficient to guarantee classification performance in the nearest neighbor search. Thus, SVM is shown to be more robust to various system configurations scalable with \(N_{t}\).  
In Figure 9, we compare the sum rate over the number of classes \(L^{\prime}\) when \(N_{t}=5\), \(U=3\) and \(L^{\prime}\in\{1,2,4,8,16,32,60\}\). It is observed that the sum rates of SVM and \(k\)-NN coincide with that of RAND when \(L^{\prime}=1\), and they increase monotonically with \(L^{\prime}\).  
To demonstrate our SVM-based approach in a massive multi-antenna setting, in Figure 10 we plot the sum rate over \(N_{t}\) from 20 to 100 when \(U=2\) and \(L^{\prime}=60\). It is observed that, unlike the results in Figure 10, the sum rate of SVM is reduced with increasing \(N_{t}\). Such a degradation occurs because the number of classes, \(L^{\prime}\), is over-reduced for large \(N_{t}\). To overcome this problem, \(L^{\prime}\) needs to scale according to the size of \(N_{t}\), which is not employed in our study, however, since such a scaling of \(L^{\prime}\) should be accompanied by a much larger training dataset, which may cause a memory overflow, in order to guarantee a sum-rate performance comparable to that of OPT. Using a sophisticated offline training method to appropriately adjust the number of classes without any memory overflow remains a goal for future work. However, it can be seen that SVM still offers substantial gains in terms of sum rate compared to RAND.  
Figure 8: Sum rate over \(N_{t}\) when \(U=3\).  
Figure 9: Sum rate over \(L^{\prime}\) when \(N_{t}=5\) and \(U=3\).