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

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 b...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Dahal, Biraj, Havrilla, Alex, Chen, Minshuo, Zhao, Tuo, Liao, Wenjing
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.02.2023
Subjects
Data structures
Datasets
Manifolds (mathematics)
Networks
Smoothness
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422