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Poster
On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds
Biraj Dahal · Alexander Havrilla · Minshuo Chen · Tuo Zhao · Wenjing Liao

Wed Nov 30 09:00 AM -- 11:00 AM (PST) @ Hall J #932
in Poster Session 3 »

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.

Keywords: distribution estimation · low-dimensional manifold · Deep generative models
[ Paper]  [ Poster]  [ OpenReview] 

Author Information

Biraj Dahal (Georgia Institute of Technology)
Alexander Havrilla (Georgia Institute of Technology)
Minshuo Chen (Princeton University)
Tuo Zhao (Georgia Tech)
Wenjing Liao (Georgia Tech)

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