A universal approximation theorem and its applications to vector lattice theory

TL;DR

This paper extends classical approximation theorems to infinite-dimensional settings, applying to neural networks and vector lattice theory, and improves existing density results and related approximation theorems.

Contribution

It presents infinite-dimensional variants of a classical approximation theorem, enhancing neural network density results and applications in vector lattice theory.

Findings

Extended density results to infinite-dimensional spaces.

Improved approximation theorems in vector lattices.

Enhanced neural network approximation capabilities.

Abstract

A classical result in approximation theory states that for any continuous function \( \varphi: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if \( \varphi \) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improve the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}.