## 3 Results
### Optimal Architectures  
To optimally adjust these finds with all three datasets, the best GAN architecture that presented no mode collapse was obtained using 3 hidden layers in the generator and two hidden layers in the discriminator. The size of hidden layers are conditioned by the size of the target feature space (Figure 4). To use the example of the largest target domain (DS3 = \(\mathbb{R}^{60}\)), the generator is programmed so that the first hidden layer (\(\mathit{Gh}^{1}\)) is a quarter of the size of the target vector (in this case \(\mathit{Gh}^{1}\) = 60\(\cdot\)1/4= 15). If this calculation produces a decimal value (e.g., 59/4 = 14.75), the ceiling of this number is taken (\(\lceil 1/4\cdot 59\rceil=15\)). This is followed by a layer half the size of the target vector (\(\mathit{Gh}^{2}\) = \(\lceil 1/2\cdot 60\rceil=30\)). The generator's final hidden layer is the size of the vector plus one quarter (\(\mathit{Gh}^{3}\) = \(\lceil 1/4\cdot 60\rceil+60=75\)). The discriminator, on the other hand, is composed of two hidden layers, with the first hidden layer being the same size as \(\mathit{Gh}^{2}\), and the second hidden layer equivalent to \(\mathit{Gh}^{1}\). Each hidden layer is followed by a batch normalization algorithm before being activated using the tanh function. Tanh works best considering the target domain is scaled to values between -1 and 1. Other components of the algorithm include a dropout layer (\(p\) = 0.4) prior to the discriminator's output and random normal kernel initializers in both models (discriminator \(\sigma\) = 0.1, generator \(\sigma\) = 0.7).  
Experiments with loss functions and optimization algorithms showed a significant improvement in performance using \(\mathit{LSGAN}\) and \(\mathit{WGAN}\) variants when compared with vanilla GAN's binary cross-entropy (rTOST \(p\) < 0.05, Table 3 with more details explained in Section 3.2). All three GANs were able to generate realistic distributions, with \(\mathit{WGAN}\)-\(\mathit{GP}\) outperforming \(\mathit{WGAN}\) in some cases (Table 3). \(\mathit{LSGAN}\) additionally worked best when using Adam optimization (\(\alpha\) = 0.0002, \(\beta_{1}\) = 0.5), while \(\mathit{WGAN}\) and \(\mathit{WGAN}\)-\(\mathit{GP}\) excelled using RMSprop (\(\alpha\) = 0.00005).  
The optimal batch number was found at 16. This allowed the discriminator enough data to objectively evaluate performance and, thus, resulted in more efficient weight updates for the generator. The number of epochs, however, were highly dependent on the number of individuals used for training.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\mathit{LSGAN}\)} & \multicolumn{2}{c}{\(\mathit{WGAN}\)} & \multicolumn{2}{c}{\(\mathit{WGAN}\)-\(\mathit{GP}\)} \\ \cline{2-7}  & \(\lvert\mathit{d}\rvert\) & \(\boldsymbol{p}\) & \(\lvert\mathit{d}\rvert\) & \(\boldsymbol{p}\) & \(\lvert\mathit{d}\rvert\) & \(\boldsymbol{p}\) \\ \hline DS1 & 0.187 & 0.0105 & 0.075 & \(2.9\times 10^{-6}\) & 0.169 & 0.0013 \\ DS2 & 0.019 & \(2.2\times 10^{-16}\) & 0.043 & \(1.3\times 10^{-11}\) & 0.002 & \(2.1\times 10^{-20}\) \\ DS3 & 0.052 & \(7.3\times 10^{-20}\) & 0.040 & \(6.7\times 10^{-23}\) & 0.031 & \(3.6\times 10^{-22}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Best obtained absolute difference and p value calculations for robust equivalency testing of each of the synthesized distributions using different GANs.  
Finally, the dimensionality of latent space was also found to be conditioned by the size of the target domain, as will be explained in continuation.