## 5 Results and Discussion  
### Nonlinear Trajectory Interpolation  
We begin with demonstrating the efficacy of the proposed Renyi GAN training methodology to learn the mapping and then interpolate between points from a population whose distribution follows a spiral pattern which is a two-dimensional variant of the Swiss roll dataset:  
\[(x_{1},x_{2}) = (y\cos{(y)},y\sin{(y)})+\varepsilon\,,\quad\text{with}\quad y= \frac{3\pi}{2}(1+2u)\,, \tag{10}\] \[u \sim \mathcal{U}(0,1)\quad\text{and}\quad\varepsilon\sim\mathcal{N}( \mathbf{0},\varepsilon^{2}I_{2})\,,\]  
where \(y\) acts as the condition variable and \(\varepsilon=0.2\). As explained in Section 2, we construct the embedding vector of the conditional variable, \(y\), to span a 10-dimensional interval with boundaries \(e_{1}=(1,\dots,1)\) and \(e_{2}=-e_{1}\). Moreover, we set the generator's noise input, \(z\), to be a 2-dimensional Gaussian vector. We found that the training steps required by the FNN generator to converge were half of those required by the GMM generator (Supplementary Figure S4). To speed up convergence for both cases, we used gradient penalty during optimization thus restricting the function space of the discriminator to Lipschitz continuous functions. We further apply the covariance penalty as explained in Section 3.2.1 for the GMM generator. We also initialize the parameters of the GMM generator using the k-means algorithm which provides a reasonable starting point and improves both acceleration and convergence stability of the training process. Moving forward, it is well-known that different divergences (here, different values for the order of Renyi divergence, \(\alpha\)) result in fundamentally different behavior during training affecting thus the obtained solution [42, 43]. For instance, KLD minimization (i.e., \(\alpha=1\)) tends to produce a distribution that covers all the modes while the reverse KLD (i.e., \(\alpha=0\)) tends to produce a distribution that often contains a subset of the modes. For the spiral-shaped distribution, we experimentally verified that reverse KLD is a more suitable choice for the loss function. Indeed, reverse KLD favors the concentration of the mass towards the mean instead of covering the tails (see Supplementary Figure S1). Figure 3 demonstrates the generative capabilities of the proposed methodology in two cases: (A) when training data for the full range of values of the conditioning variable exist and (B) when no training data exist for two distant intervals of the conditioning variable. In particular, the left panel of Figure 3A shows training samples with the condition, \(y\), encoding with coloring of the samples and being scaled in the interval \([0,1]\). Accordingly, the left panel of Figure 3B shows the training samples for \(y\in[0,0.25]\cup[0.3,0.6]\cup[0.65,1]\). The middle and right panels of Figure 3 show the generated data after training using the conditional FNN and the conditional GMM generator, respectively. The comparison between Figure 3A,B reveals that training on the complete data set is more accurate and also faster (see Supplementary Figure S4). Furthermore, there is a trade-off between accuracy and interpretability as in both cases the reconstruction achieved by the conditional FNN variant is relatively smoother than that of the conditional GMM generator. This trade-off emerges not only as a result of the neural network's non-linear activation function but also as a consequence of the capacity of each generator. A ZIMM (or GMM) generator is essentially a shallow, one-layer model while FNNs can be constructed arbitrarily deep, in principle. Regardless, both generator variants manage to learn to a great extent the real distribution and delineate the original population trajectory. Interestingly, when the GMM generator is used the generated distribution resembles a piecewise linear interpolation curve between key data points on the original trajectory.