Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem

TL;DR

This paper establishes error bounds for deep ReLU networks approximating multivariate functions, leveraging the Kolmogorov--Arnold superposition theorem to reduce the curse of dimensionality.

Contribution

It provides a constructive proof linking the Kolmogorov--Arnold theorem to deep ReLU network approximation, highlighting a subset of functions efficiently approximable.

Findings

Reduced curse of dimensionality in approximation error bounds

Constructive method for approximating multivariate functions

Identification of function subsets suitable for deep ReLU approximation

Abstract

We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened. Our theorem is based on a constructive proof of the Kolmogorov--Arnold superposition theorem, and on a subset of multivariate continuous functions whose outer superposition functions can be efficiently approximated by deep ReLU networks.