Learning disconnected manifolds: a no GANs land
This addresses a fundamental limitation in generative modeling for researchers and practitioners, offering a solution to improve performance on datasets with disconnected modes, though it is incremental as it builds on existing GAN frameworks.
The paper tackles the problem of Generative Adversarial Networks (GANs) struggling to learn disconnected manifolds due to their continuous generators, establishing a no free lunch theorem that sets an upper bound on precision. It proposes a rejection sampling method based on the generator's Jacobian norm and demonstrates its effectiveness on models like BigGAN.
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.