A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributions
It addresses the theoretical expressiveness of neural networks for distribution approximation, which is foundational for generative modeling and statistical learning, but is incremental as it builds on existing universal approximation theorems.
This paper proves that deep neural networks can universally approximate probability distributions by mapping a source distribution to a target distribution via the gradient of a ReLU network, with error bounds measured by Wasserstein distance, MMD, and KSD, showing exponential network size growth in dimension for Wasserstein but polynomial for MMD and KSD.
This paper studies the universal approximation property of deep neural networks for representing probability distributions. Given a target distribution $π$ 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 push-forward measure $(\nabla g)_\# p_z$ of $p_z$ under the map $\nabla g$ is arbitrarily close to the target measure $π$. 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 semi-discrete optimal transport.