Nonlinear Approximation and (Deep) ReLU Networks

TL;DR

This paper investigates the approximation capabilities of deep ReLU neural networks for univariate functions, demonstrating they can efficiently approximate classes of functions that challenge traditional nonlinear approximation methods.

Contribution

It establishes that neural networks possess greater approximation power than classical nonlinear methods by identifying function classes they can efficiently capture.

Findings

Neural networks outperform traditional nonlinear approximation methods on certain function classes.

ReLU networks can efficiently approximate functions that are difficult for classical methods.

The paper provides theoretical insights into the approximation power of neural networks for univariate functions.

Abstract

This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approximate functions with classical smoothness to the same accuracy as classical linear methods of approximation, e.g. approximation by polynomials or by piecewise polynomials on prescribed partitions. However, approximation by neural networks depending on n parameters is a form of nonlinear approximation and as such should be compared with other nonlinear methods such as variable knot splines or n-term approximation from dictionaries. The performance of neural networks in targeted applications such as machine learning indicate that they actually possess even greater approximation power than these traditional methods of nonlinear approximation. The main results of this article prove that this is indeed the case. This is done by exhibiting large classes of functions which can be efficiently captured by neural networks where classical nonlinear methods fall short of the task. The present article purposefully limits itself to studying the approximation of univariate functions by ReLU networks. Many generalizations to functions of several variables and other activation functions can be envisioned. However, even in this simplest of settings considered here, a theory that completely quantifies the approximation power of neural networks is still lacking.