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A universal approximation theorem and its applications to vector lattice theory

Eugene Bilokopytov, Foivos Xanthos

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}.

A universal approximation theorem and its applications to vector lattice theory

Abstract

A classical result in approximation theory states that for any continuous function , the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if 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}.

Paper Structure

This paper contains 8 sections, 23 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. A Universal approximation theorem and its extensions
  3. The scalar case
  4. Infinite dimensional variants
  5. Applications to vector lattice theory
  6. Relatively uniformly closed sublattice generated by a set
  7. Other applications
  8. Other approximation results

Key Result

Theorem 1.1

Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous sigmoid function. Then $\mathfrak{N}(\sigma_\phi)$ is dense in $C(\mathbb{R}^m)$.

Theorems & Definitions (42)

  • Theorem 1.1: Sigmoidal-UAT
  • Theorem 1.2: Non-polynomial-UAT
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['lybenko']}
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • ...and 32 more