# Genetic Algorithms to Maximize the Relevant Mutual Information in Communication Receivers +
## 4 Results and Discussion
### Quantization of the Channel Output with Minimum Loss of Relevant Information
#### 4.1.3 Genetic Algorithm Quantizer Design with Approximate Nearest Neighbor Search  
The numbers presented in the prior section illustrate that for \(|\mathcal{T}|=16\), there is still a mentionable gap between \(\mathrm{I}(\mathsf{X};\mathsf{Y})\) and \(\mathrm{I}(\mathsf{X};\mathsf{T})\) for all considered quantizers. In order to close that gap, one has to increase the output cardinality \(|\mathcal{T}|\) of the quantizer to further decrease the quantization loss. This, however, proportionally increases the number of distance calculations for the method from Section 3.1.1. Here we use the approximate \begin{table}
\begin{tabular}{c l c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(n_{\mathrm{pop}}\) & population size & 200 \\ \(n_{\mathrm{cross}}\) & number of crossovers for genetic combination & 5 \\ \(p_{\mathrm{mut}}\) & mutation probability of each bit in the genotype & \(10^{-4}\) \\ \(r_{m}\) & number of bits to represent a parameter in the genotype & 10 \\ \(n_{\mathrm{elite}}\) & number of elite population members (guaranteed to breed) & 20 \\ \(n_{\mathrm{gen}}\) & number of evolved generations & 2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of parameters of the genetic algorithm. Figure 7: Quantization regions for the output of an AWGN channel with 8-PSK modulation in the complex plane for \(\sigma_{n}^{2}=0.5\), \(|\mathcal{T}|=16\) constructed with the genetic algorithm. nearest neighbor search algorithm from Section 3.1.3 to overcome that issue. Figure 8 compares the preserved relevant information \(\mathrm{I}(\mathrm{X};\mathrm{T})\) of the KL-means algorithm and the proposed genetic algorithm optimized compression mappings for an output cardinality of \(|\mathcal{T}|=256\) as a function of the SNR \(1/\sigma_{n}^{2}\) of the AWGN channel. Due to its simpler distance calculation, we only consider the Manhattan distance \(d_{\mathrm{M}}(\mathbf{y},\boldsymbol{\theta}_{t})\) here. For this investigation, we were forced to decrease the cardinality of the grid that finely discretizes the complex plane to \(\sqrt{|\mathcal{Y}|}=128\) points for \(y^{\mathrm{re}},y^{\mathrm{im}}\in[-1.5,1.5]\), respectively. The reason is that the time complexity of the KL-means algorithm from [26] is proportional to the product \(|\mathcal{T}|\cdot|\mathcal{Y}|\) and with \(|\mathcal{Y}|=65,536\) as used in the previous investigation and \(|\mathcal{T}|=256\) used here, it was just not possible to create the KL-means quantizers in a reasonable time, even though we have used a highly-parallel implementation of that algorithm which parallelizes the algorithm on a graphics card [32]. Please note that using a coarser grid slightly degrades \(\mathrm{I}(\mathrm{X};\mathrm{Y})\). This indicates that using the KL-means algorithm for very large cardinalities \(|\mathcal{T}|\) is challenging. The method proposed here, however, easily allows using such a large \(|\mathcal{T}|\). The approximate nearest neighbor search algorithm from Section 3.1.3 used the parameters \(n_{\mathrm{neighbor}}=6\) neighbors, \(n_{\mathrm{entry}}=6\) entry nodes and a maximum path length of \(l_{\mathrm{path}}^{\mathrm{max}}=5\). These parameters were found to offer a good tradeoff between sparsity of the neighborhood graph and performance. The worst case number of distance calculations to determine \(t\) for a given \(\mathbf{y}\) in this setting is \(n_{\mathrm{dist}}^{\mathrm{max}}=6+6\cdot 5=36\) according to Equation (6) which is significantly less than \(|\mathcal{T}|=256\). During our experiments we found out that it is even possible to reduce the maximum number of distance calculations further by decreasing \(n_{\mathrm{entry}}\), \(n_{\mathrm{neighbor}}\) or \(l_{\mathrm{path}}^{\mathrm{max}}\) at the expense of very slight losses in \(\mathrm{I}(\mathrm{X};\mathrm{T})\). Moreover, we have added the choice of the first entry node as a parameter to the genetic algorithm such that it is included in the optimization process.