## 3 Results  
All three datasets analyzed present highly inhomogeneous multivariate distributions (\(p<2.2\times 10^{-16}\)). Univariate comparisons (Table 2), however, present a mixture of both inhomogeneous and homogeneous distributions across PC1 and PC2, where the majority of variance is represented.  
GAN failure through mode collapse was frequently observed throughout most of the initial trials, characterized by an intense clustering of points with little to no variation in feature space (Figure 3). Qualitatively, this type of failure is easily diagnosed by visual inspection of graphs. Quantitatively, mode failure can be characterized by a dramatic decrease in variance seen through deviation metrics. To provide an example, Figure 1 presents the use of a vanilla GAN trained on DS2. At first, training can be seen to start well with the closest (yet not optimal) approximation to the target domain's median (Figure 3A). Nevertheless, little variation is present (\(BWMV\) of PC1 = 0.14, target \(BWMV=0.38\)). As training continues, the algorithm is unable to find the correct median, and performance is seen to deteriorate (Figure 3B). This presents an exponential decrease in the variance of synthetic data (Figure 3B PC1 \(BWMV\) = 0.02, Figure 3C PC1 \(BWMV\) = 0.0002). Likewise, through mode collapse, the generator is unable to map the true normality of the distribution, generating increasingly normal data in PC1 (Shapiro \(w>0.98\), \(p>0.56\)).  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{2}{c}{**PC1**} & \multicolumn{2}{c}{**PC2**} \\ \cline{3-6}  & \multicolumn{1}{c}{**Domain**} & \multicolumn{1}{c}{**PCs with 95\%**} & \multicolumn{1}{c}{**Variance**} & \multicolumn{1}{c}{**Shapiro Test w**} & \multicolumn{1}{c}{**Variance**} & \multicolumn{1}{c}{**Shapiro Test w**} \\ \cline{3-6}  & \multicolumn{1}{c}{**Dimensionality**} & \multicolumn{1}{c}{**Cumulative Variance**} & \multicolumn{1}{c}{**(\%)**} & \multicolumn{1}{c}{**(\(p\))**} & \multicolumn{1}{c}{**(\%)**} & \multicolumn{1}{c}{**(\(p\))**} \\ \hline DS1 & \(\mathbb{R}\)14 & 4 & 69.92 & 0.95 (0.02) & 14.37 & 0.97 (0.15) \\ DS2 & \(\mathbb{R}\)39 & 11 & 32.27 & 0.96 (0.05) & 25.70 & 0.98 (0.31) \\ DS3 & \(\mathbb{R}\)60 & 13 & 32.83 & 0.99 (0.75) & 19.55 & 0.98 (0.30) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of each dataset’s target domain with univariate calculations of distribution normality in the top two PC scores.  
Figure 3: Examples of GAN failure in the form of (**A**) slight, (**B**) large and (**C**) extreme mode collapse when used to augment DS2.  
Replacing Leaky ReLU with tanh activation functions resulted in a significant improvement of generated sample medians (difference in median for Leaky ReLU = 0.78; tanh = 0.26), yet with little improvement in \(\mathit{BWMV}\).  
To overcome mode collapse, kernel initializers and batch normalization algorithms were incorporated into both the generator and the discriminator. Batch normalization was included before activation, presenting an increase in \(\mathit{BWMV}\). Initializers required careful adjustment, with small standard deviation values resulting in mode collapse. Additional experiments found the discriminator to require a more intense initializer (\(\sigma\) = 0.1) than the generator (\(\sigma\) = 0.7), while optimal results were obtained using a random normal distribution. Such a configuration allows the generator more room to adjust its weights, finding the best way of reaching the target domain's median and absolute deviation while preventing the discriminator from learning too quickly.  
Experiments adjusting hidden layer densities found symmetry between the generator and the discriminator to be unnecessary. The generator was seen to require more hidden layers in order to learn the distributions efficiently, while a larger density than the output in the last hidden layer also produced an increase in performance. The discriminator worked best with just two hidden layers.