Neural Network Approximation of Refinable Functions

TL;DR

This paper demonstrates that refinable functions, used in wavelets and geometric design, can be efficiently approximated by deep ReLU neural networks with fixed width and increasing depth, achieving exponential accuracy.

Contribution

It extends neural network approximation theory to include refinable functions, showing they can be approximated with exponential accuracy by deep ReLU networks.

Findings

Refinable functions are approximated with exponential accuracy by neural networks.

Results apply to wavelet functions and subdivision algorithms in geometric design.

Deep ReLU networks with fixed width and increasing depth are effective for these functions.

Abstract

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions (see e.g. \cite{E,S,Y}) and, on the other hand, very rough functions such as the Weierstrass function (see e.g. \cite{EPGB,DDFHP}), which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.