Computation complexity of deep ReLU neural networks in high-dimensional approximation

TL;DR

This paper analyzes the computational complexity of deep ReLU neural networks in high-dimensional function approximation, providing explicit bounds and demonstrating the benefits of adaptive methods over nonadaptive ones.

Contribution

It introduces explicit constructions and bounds for deep ReLU networks approximating functions in high-dimensional spaces, highlighting the advantages of adaptive approximation methods.

Findings

Adaptive neural networks outperform nonadaptive ones in high-dimensional approximation.

Explicit bounds on network size and depth depend on dimension and accuracy.

Constructive methods for approximating functions in H"older-Nikol'skii spaces.

Abstract

The purpose of the present paper is to study the computation complexity of deep ReLU neural networks to approximate functions in H\"older-Nikol'skii spaces of mixed smoothness $H_\infty^\alpha(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in H_\infty^\alpha(\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive deep ReLU neural networks having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove dimension-dependent bounds for the computation complexity of this approximation, characterized by the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. Our results show the advantage of the adaptive method of approximation by deep ReLU neural networks over nonadaptive one.