A note on Cybenko's Universal Approximation Theorem

TL;DR

This paper identifies a critical flaw in Cybenko's proof of the universal approximation theorem, raising questions about the validity of the theorem and its measure-theoretic implications.

Contribution

It highlights a mistake in a widely cited proof, prompting a re-evaluation of the theorem's validity and inspiring further research in measure theory.

Findings

The proof contains a significant error affecting the theorem's validity

The mistake suggests the need for alternative proof strategies

Raises new questions in measure theory related to neural network approximation

Abstract

In this short note, we point out a mistake in G.Cybenko's proof of his version of the universal approximation theorem which has been widely cited. This mistake might not be easily fixable along the idea of his proof and it also leads to an interesting question in measure theory.