High-Dimensional Distribution Generation Through Deep Neural Networks

TL;DR

This paper demonstrates that deep neural networks can generate any bounded $d$-dimensional distribution from a 1-dimensional uniform input without approximation error, emphasizing the importance of network depth and linking to quantization limits.

Contribution

It generalizes space-filling approaches to show deep networks can perfectly generate high-dimensional distributions from low-dimensional inputs, highlighting depth's role.

Findings

Deep networks can generate any bounded $d$-dimensional distribution from 1D input.

Network depth is crucial for minimizing Wasserstein distance in distribution approximation.

Encoding complexity for histogram distributions reaches fundamental quantization limits.

Abstract

We show that every $d$-dimensional probability distribution of bounded support can be generated through deep ReLU networks out of a $1$-dimensional uniform input distribution. What is more, this is possible without incurring a cost - in terms of approximation error measured in Wasserstein-distance - relative to generating the $d$-dimensional target distribution from $d$ independent random variables. This is enabled by a vast generalization of the space-filling approach discovered in (Bailey & Telgarsky, 2018). The construction we propose elicits the importance of network depth in driving the Wasserstein distance between the target distribution and its neural network approximation to zero. Finally, we find that, for histogram target distributions, the number of bits needed to encode the corresponding generative network equals the fundamental limit for encoding probability distributions as dictated by quantization theory.