A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributions
Abstract
This paper studies the universal approximation property of deep neural networks for representing probability distributions. Given a target distribution $\pi$ and a source distribution $p_z$ both defined on $\mathbb{R}^d$, we prove under some assumptions that there exists a deep neural network $g:\mathbb{R}^d\rightarrow \mathbb{R}$ with ReLU activation such that the pushforward measure $(\nabla g)_\# p_z$ of $p_z$ under the map $\nabla g$ is arbitrarily close to the target measure $\pi$. The closeness are measured by three classes of integral probability metrics between probability distributions: $1$Wasserstein distance, maximum mean distance (MMD) and kernelized Stein discrepancy (KSD). We prove upper bounds for the size (width and depth) of the deep neural network in terms of the dimension $d$ and the approximation error $\varepsilon$ with respect to the three discrepancies. In particular, the size of neural network can grow exponentially in $d$ when $1$Wasserstein distance is used as the discrepancy, whereas for both MMD and KSD the size of neural network only depends on $d$ at most polynomially. Our proof relies on convergence estimates of empirical measures under aforementioned discrepancies and semidiscrete optimal transport.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 DOI:
 10.48550/arXiv.2004.08867
 arXiv:
 arXiv:2004.08867
 Bibcode:
 2020arXiv200408867L
 Keywords:

 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning
 EPrint:
 Accepted in the Thirtyfourth Conference on Neural Information Processing Systems (NeurIPS 2020)