  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.
