On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds

TL;DR

This paper provides theoretical guarantees for deep generative models assuming data lies on a low-dimensional manifold, showing convergence rates depend on intrinsic rather than ambient dimension, thus justifying their empirical success.

Contribution

The paper proves statistical guarantees for generative networks under Wasserstein-1 loss with data on low-dimensional manifolds, without smoothness assumptions.

Findings

Wasserstein-1 loss converges at a rate depending on intrinsic dimension.

Generative networks are justified by theory under manifold assumptions.

No smoothness assumptions are needed on data distribution.

Abstract

Generative networks have experienced great empirical successes in distribution learning. Many existing experiments have demonstrated that generative networks can generate high-dimensional complex data from a low-dimensional easy-to-sample distribution. However, this phenomenon can not be justified by existing theories. The widely held manifold hypothesis speculates that real-world data sets, such as natural images and signals, exhibit low-dimensional geometric structures. In this paper, we take such low-dimensional data structures into consideration by assuming that data distributions are supported on a low-dimensional manifold. We prove statistical guarantees of generative networks under the Wasserstein-1 loss. We show that the Wasserstein-1 loss converges to zero at a fast rate depending on the intrinsic dimension instead of the ambient data dimension. Our theory leverages the low-dimensional geometric structures in data sets and justifies the practical power of generative networks. We require no smoothness assumptions on the data distribution which is desirable in practice.