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

Biraj Dahal, Alexander Havrilla, Minshuo Chen, Tuo Zhao, Wenjing Liao

Keywords: Deep generative models, distribution estimation, low-dimensional manifold

TL;DR: We prove approximation and statistical estimation theories of deep generative models for distribution learning when the distribution is supported on a low-dimensional manifold.

Abstract: Deep generative models have experienced great empirical successes in distribution learning. Many existing experiments have demonstrated that deep generative networks can efficiently 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 approximation and estimation theories of deep generative networks for estimating distributions on a low-dimensional manifold 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 deep generative models. We require no smoothness assumptions on the data distribution which is desirable in practice.