On the optimal precision of GANs

Thibaut Issenhuth, Ugo Tanielian, Jeremie Mary, David Picard

Keywords: Deep learning theory, GANs, Deep generative modelling

Abstract: Generative adversarial networks (GANs) are known to face model misspecification when learning disconnected distributions. Indeed, continuous mapping from a unimodal latent distribution to a disconnected one is impossible, so GANs necessarily generate samples outside of the support of the target distribution. In this paper, we make the connection between the performance of GANs and their latent space configuration. In particular, we raise the following question: what is the latent space partition that minimizes the measure of out-of-manifold samples? Building on a recent result of geometric measure theory, we prove there exist optimal GANs when the dimension of the latent space is larger than the number of modes. In particular, we show that these generators structure their latent space as a `simplicial cluster' - a Voronoi partition where centers are equally distant. We derive both an upper and a lower bound on the optimal precision of GANs learning disconnected manifolds. Interestingly, these two bounds have the same order of decrease: $\sqrt{\log m}$, $m$ being the number of modes. Finally, we perform several experiments to exhibit the geometry of the latent space and experimentally show that GANs have a geometry with similar properties to the theoretical one.

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