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Many generative models synthesize data by transforming a standard Gaussianrandom variable using a deterministic neural network. Among these models arethe Variational Autoencoders and the Generative Adversarial Networks. In thiswork, we call them "push-forward" models and study their expressivity. We showthat the Lipschitz constant of these generative networks has to be large inorder to fit multimodal distributions. More precisely, we show that the totalvariation distance and the Kullback-Leibler divergence between the generatedand the data distribution are bounded from below by a constant depending on themode separation and the Lipschitz constant. Since constraining the Lipschitzconstants of neural networks is a common way to stabilize generative models,there is a provable trade-off between the ability of push-forward models toapproximate multimodal distributions and the stability of their training. Wevalidate our findings on one-dimensional and image datasets and empiricallyshow that generative models consisting of stacked networks with stochasticinput at each step, such as diffusion models do not suffer of such limitations.

Can Push-forward Generative Models Fit Multimodal Distributions?