Abstract

Natural data is often constrained to an low-dimensional manifold. This work focuses on the task of building theoretically principled generative models for such data. Current generative models learn the manifold by mapping an low-dimensional latent variable through a neural network. These procedures, which we call pushforward models, incur a straightforward limitation: manifolds cannot in general be represented with a single parameterization, meaning that attempts to do so will incur either computational instability or the inability to learn probability densities within the manifold. To remedy this problem, we propose to model the manifold as a neural implicit manifold: the set of zeros of a neural network. We then learn the probability density within the manifold with a constrained energy-based model, which employs a constrained variant of Langevin dynamics to train and sample from the learned manifold. In experiments on synthetic and natural data, we show that our model can learn manifold-supported distributions with complex topologies more accurately than pushforward models.