# Deep Network Approximation Characterized by Number of Neurons

## TL;DR

This paper provides a detailed quantitative analysis of how the approximation capabilities of deep ReLU neural networks depend on the number of neurons, depth, and the domain's geometry, including irregular and low-dimensional manifolds.

## Contribution

It offers a constructive characterization of the approximation rates of deep neural networks for various classes of functions and domains, extending previous results to irregular and low-dimensional manifolds.

## Key findings

- ReLU FNNs can approximate Hölder continuous functions with nearly optimal rates.
- Approximation rates depend on the number of neurons, depth, and domain complexity.
- Extension of approximation analysis to irregular domains and low-dimensional manifolds.

## Abstract

This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+1\}\big)$ and depth $\mathcal{O}(L)$ can approximate an arbitrary H\"older continuous function of order $\alpha\in (0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}\big(\sqrt{d} N^{-2\alpha/d}L^{-2\alpha/d}\big)$ measured in $L^p$-norm for any $N,L\in \mathbb{N}^+$ and $p\in[1,\infty]$. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $\omega_f(\cdot)$, the constructive approximation rate is $\mathcal{O}\big(\sqrt{d}\,\omega_f( N^{-2/d}L^{-2/d})\big)$. We also extend our analysis to $f$ on irregular domains or those localized in an $\varepsilon$-neighborhood of a $d_{\mathcal{M}}$-dimensional smooth manifold $\mathcal{M}\subseteq [0,1]^d$ with $d_{\mathcal{M}}\ll d$. Especially, in the case of an essentially low-dimensional domain, we show an approximation rate $\mathcal{O}\big(\omega_f(\tfrac{\varepsilon}{1-\delta}\sqrt{\tfrac{d}{d_\delta}}+\varepsilon)+\sqrt{d}\,\omega_f(\tfrac{\sqrt{d}}{(1-\delta)\sqrt{d_\delta}}N^{-2/d_\delta}L^{-2/d_\delta})\big)$ for ReLU FNNs to approximate $f$ in the $\varepsilon$-neighborhood, where $d_\delta=\mathcal{O}\big(d_{\mathcal{M}}\tfrac{\ln (d/\delta)}{\delta^2}\big)$ for any $\delta\in(0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_{\delta}$-dimensional domain.