## 5 Results and Discussion
### Single Cell Data Augmentation  
Finally, we test the training efficacy of our methodology to learn distributions in real biological settings and particularly when the number of available observations from a given cell population is low. This is a fundamental problem in life sciences for which an artificial data augmentation has been recently demonstrated to be a realistic solution [21]. To this direction, we used real measurements from [45] (data were accessed from [[https://community.cytobank.org/cytobank/experiments/46098/illustrations/121588](https://community.cytobank.org/cytobank/experiments/46098/illustrations/121588)]([https://community.cytobank.org/cytobank/experiments/46098/illustrations/121588](https://community.cytobank.org/cytobank/experiments/46098/illustrations/121588)), accessed on 15 April 2020). Specifically, we consider single cell mass cytometry measurements \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Generator** & \(\mathbf{y\in[0,1]}\) & \(\mathbf{y\in[0.4,0.6]}\) & \(\mathbf{y\in\text{State 2}}\) \\ \hline
**cFNN** & 0.0748 & 0.2308 & 0.2430 \\
**cZIMM** & 0.0763 & 0.5090 & 0.1719 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MMD computations using FNN and ZIMM for \(\sigma=5\) in the RBF kernel, for experiments in Figure 4. Reference is the average MMD over the test set and 50 randomly chosen equally-sized training sets and is equal to & \(0.015\pm 0.005\). Lower MMD value is preferable. Figure 4: Synthetic 50-dimensional data set with zero-inflated modes. First two PCA components visualization of real over generated data, colored in turquoise and orange respectively. (**upper row**) Training data with uniformly chosen labels \(y\in[0,1]\). Conditional FNN shown in the middle is a better fit in comparison to the conditional ZIMM model on the right, at the cost of interpretability. Detailed histograms over all 50 genes (distribution marginals), can be found in the Supplementary Figure S13. (**middle row**) Training data colored according to pseudo-time label \(y\) (continuous) is shown on the left, where all five intervals have similar sample sizes. Training is performed on data with condition \(y\in[0,0.4]\cup[0.6,1.0]\) (turquoise) whereas we generate data with \(y\in[0.4,0.6]\) (orange). Interpolation to unseen data is more accurate when the conditional FNN generator is used though conditional ZIMM is capable of identifying a significant part of the test set. (**lower row**) Training data colored according to differentiation state label \(y\) (discrete) is shown on the left. Conditional FNN (middle) and conditional ZIMM (right), using a subpopulation of 5% of samples from state 2. The latter is important because this discrete condition does not provide dynamical information for interpolation. on 16 bone marrow protein markers (\(d=16\)) coming from healthy and disease-carrying individuals. In total, the data set consists of almost 200K healthy and disease-related cell samples. Before analysis, data were transformed using the inverse hyperbolic sine \(arcsinh()\) transformation with a cofactor of 5, which is typical in order to have comparable supports across dimensions. To simulate the existence of a rare cell subpopulation in the data we proceed as follows. First, we collect 26K random cell samples from the healthy population. Then, we consider a case where the number of cells from the rare population is 2% of that of the healthy one and another where it is only 1%. Our goal is to train the proposed conditional generator variants on distributions of rare cell population and then use them to generate new realistic samples for data augmentation. Figure 5 shows the effectiveness of each variant when the training data consist of 2% or 1% of the disease-related subpopulation of samples. In general, we see that both FNN (left) and ZIMM (right) conditional variants are able to generate realistic samples.