Deep Network Approximation Characterized by Number of Neurons

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 $\omegaf(\cdot)$, the constructive approximation rate is $\mathcal{O}\big(\sqrt{d}\,\omegaf( N^{{-2/d}L{-2/d})\big)$.} 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 $\mathcal{O}\big(\omegaf(\tfrac{\varepsilon}{1-\delta}\sqrt{\tfrac{d}{d\delta}}+\varepsilon)+\sqrt{d}\,\omegaf(\tfrac{\sqrt{d}}{(1-\delta)\sqrt{d\delta}}N^{{-2/d\delta}L{-2/d\delta})\big)$} 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.