Typical architectures of Generative AdversarialNetworks make use of a unimodal latent distribution transformed by a continuous generator. Consequently, the modeled distribution always has connected support which is cumbersome when learning a disconnected set of manifolds. We formalize this problem by establishing a no free lunch theorem for the disconnected manifold learning stating an upper bound on the precision of the targeted distribution. This is done by building on the necessary existence of a low-quality region where the generator continuously samples data between two disconnected modes. Finally, we derive a rejection sampling method based on the norm of generators Jacobian and show its efficiency on several generators including BigGAN.
@article{arxiv.2006.04596,
title = {Learning disconnected manifolds: a no GANs land},
author = {Ugo Tanielian and Thibaut Issenhuth and Elvis Dohmatob and Jeremie Mary},
journal= {arXiv preprint arXiv:2006.04596},
year = {2020}
}