On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds
This provides a foundational theoretical framework for generative modeling, addressing a key gap in understanding for researchers and practitioners in machine learning.
The paper tackles the theoretical justification for generative networks' empirical success in learning distributions on low-dimensional manifolds, proving that the Wasserstein-1 loss converges to zero at a fast rate dependent on intrinsic rather than ambient dimension.
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.