## 5 Results and Discussion
### Synthetic RNA-seq Data  
Next, we demonstrate the capabilities of the proposed methodology in capturing distributional changes of high dimensional populations. For this, we created a synthetic example of a population of cells that transit over three consecutive states of differentiation. We used the dyngen simulator [44] to generate \(N=50\),\(000\) samples of cells, each associated with 50 gene expression values at different timepoints \(y\) across the progression of differentiation. By design, the generated data are zero-inflated which is a defining characteristic of the respective single-cell RNAseq technology. For this simulated example we used batch size of 512 samples in order to refrain from missing out distribution regions having low mass. In addition, the dimension of the random noise input to generator was set to 18; however, we found that this parameter does not affect the quality of results. As in the previous example, we initialize the parameters related to the location of the modes of the conditional ZIMM with k-means (over training data) in order to avoid a potentially high number of training steps and/or the optimization process getting trapped in local minima. We note here that \(K=20\) zero-inflated modes are required for accurate description of the 50-dimensional distribution (see Supplementary Figure S10). Figure 4 visualizes after dimensionality reduction the performance of the conditional FNN and the interpretable conditional ZIMM on the simulated data. Each row corresponds to a different sampling scheme. Specifically, the top row shows the generated populations when the training data are uniformly sampled across the trajectory delineated by the conditioning variable \(y\). In contrast, the middle and bottom rows illustrate the performance of both variants when some interval of the conditioning variable is missing entirely. In the former case this entails the suspension of a continuous subset of the conditioning variable while, in the latter case, the suspension of a whole differentiation state, i.e., a discrete subpopulation. Evidently, in the case where all training information is available Figure 4 (top) or when a continuous subset is missing Figure 4 (middle) the conditional FNN managed to generate data that are closer to the real distribution. This is quantitatively verified by the MMD calculations reported in Table 1. Both generator models were able to interpolate between the missing parts of the original distribution. We note here that similar performances are observed if data from the interval \(y\in[0.6,0.8]\) are missing instead (Supplementary Figure S11). Supplementary Figures S5-S8 display how well each variant captures the original data in the 50-dimensional space. In contrast, when a whole differentiation state is missing from the training data both methods failed to converge (data not shown). This is because in this discrete case the conditioning \(y\) does not carry any information that relates states 1 and 3 rendering them independent. For this reason, we repeated the experiment after adding a small percentage of samples from cells at state 2. Figure 4 (bottom) shows results when 5% of these cells (or 736 in absolute numbers) have been added. As seen although both variants manage to create data similar to those coming from the real distribution, they still cannot correctly represent the real distribution of state 2. For example, the variance of the fake distribution generated using the conditional FNN is high. This is probably because the number of samples added from the missing subpopulation is not enough. In fact, we found that the variance of the generated data is positively correlated to the number of samples from the missing distribution (see Supplementary Figure S12). On the other hand, in the case of the conditional ZIMM the generated data do not create a single subpopulation that interpolates well between states 1 and 3.