## 5 Results and Discussion
### Comparison against the State-of-the-Art  
We compare our approach against another conditional GAN approach tailored for single-cell data called conditional single-cell GAN (cscGAN) [21]. Briefly, cscGAN assumes single-cell RNAseq data as input and is able to augment under-represented classes by generating realistic synthetic samples. Similarly to the cFNN and cZIMM, it is based on  
Figure 5: Two dimensional PCA plots on the real 16-dimensional data set. Healthy (turquoise) and diseased (red) cell distributions, each consisting of \(N=26\),000 samples, are shown in each subplot as reference. Cross symbols indicate the generated cell distribution (orange). **(top)** Performance of cFNN (left) and cZIMM (right) when the training set comprises of 2% (\(N=520\)) diseased samples. **(bottom)** Performance of cFNN (left) and cZIMM (right) when the training set comprises of 1% (\(N=260\)) diseased samples. Detailed histograms are included in Supplementary Figure S13. the assumption that the training samples lie on the same low-dimensional manifold [46]. However, the cscGAN approach cannot handle continuous conditioning variables while both cFNN and cZIMM are via label embedding. We trained cscGAN on the datasets of Sections 5.2 and 5.3. To achieve a fair comparison, all experiments were conducted on the basis of count data because the cscGAN architecture follows a custom normalization procedure per cell (library size normalization). Please remember that we worked with count data in the scRNASeq dataset (Section 5.2) while we performed an arcsinh transformation to the protein data in Section 5.3. Furthermore, both cscGAN generator's and discriminator's capacity was reduced because of overfitting concerns. Essentially, we reduced the number of nodes for each of the three layers by a factor between 4 and 16 without affecting the quality of the generated samples in the training set. Obviously, the downsizing of the neural networks also resulted in much faster training. Let us also remark that the respective cFNN presented in Section 5.2 is still quite smaller than the chosen cscGAN. On the scRNASeq data, we found that cscGAN fails to generate realistic samples when the under-represented distribution interpolates between consecutive cell states (see Supplementary Figure S14). This result might stem from the fact that the cscGAN approach cannot efficiently handle correlated conditioning variables in cases of low sample size. On the other hand, cscGAN produced marginally better results on the mass cytometry data. Table 2 summarizes the MMD scores when the training data consist of two separate cell populations; one abundant and another that is arbitrarily small i.e., 2% and 1% relative to the abundant one. As expected, cZIMM's MMD had the highest value. In contrast, the MMD scores of cFNN and cscGAN are on par for the 2% case while cscGAN has lower MMD for the 1% case. Accordingly, Figure 6 compares the results of our cFNN model (orange crosses) to the cscGAN approach (green crosses). As it is clearly seen, both cscGAN and cFNN are capable of producing realistic samples from the under-represented training class. However, cFNN requires significantly less training time mainly because of the smaller size of its generator. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Generator** & **2\% Subpopulation** & **1\% Subpopulation** \\ \hline
**cFNN** & 0.0302 & 0.0359 \\
**cZIMM** & 0.0351 & 0.0374 \\
**cscGAN** & 0.0296 & 0.0328 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: cFNN, cZIMM and cscGAN MMD scores for the cases of 2% and 1% rare population (RBF kernel, \(\sigma=500\)). Notice that the value of \(\sigma\) is now two orders of magnitude larger than it was in Table 3 because the data account for gene counts (i.e., unnormalized). The baseline MMD score calculated over the test set and 50 randomly chosen equally-sized training sets is equal to & \(0.0086\pm 0.002\). Lower MMD value is preferable.