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On the universal approximation of real functions with varying domain

W. Jung, C. A. Morales, L. T. T. Tran

TL;DR

This work addresses universal approximation for real-valued functions when domains vary across compact metric spaces, by leveraging the $C^0$-Gromov-Hausdorff distance $d_{GH^0}$. It extends Cybenko's one-hidden-layer network results to a varying-domain setting using a framework of locally sliceable fibrewise sets and a domain map $D$, establishing sufficient conditions under which shallow networks $N_\sigma(F)$ become dense in $\mathcal{C}=\bigcup_{X\in\mathcal{M}}\mathcal{C}(X)$. The main contribution is a density theorem (with a corollary for finite-domain families $\mathcal{F}_{fin}$) for sigmoidal activations, grounded in the discriminatory property of $\sigma$ with respect to $F$ and the separation of $B(X)$ by $F$. The results enable representations of shapes and other data via finite metric-space approximations (point clouds) by real-valued functions, providing a principled universal-approximation mechanism across variable domains.

Abstract

We establish sufficient conditions for the density of shallow neural networks \cite{C89} on the family of continuous real functions defined on a compact metric space, taking into account variations in the function domains. For this we use the Gromov-Hausdorff distance defined in \cite{5G}.

On the universal approximation of real functions with varying domain

TL;DR

This work addresses universal approximation for real-valued functions when domains vary across compact metric spaces, by leveraging the -Gromov-Hausdorff distance . It extends Cybenko's one-hidden-layer network results to a varying-domain setting using a framework of locally sliceable fibrewise sets and a domain map , establishing sufficient conditions under which shallow networks become dense in . The main contribution is a density theorem (with a corollary for finite-domain families ) for sigmoidal activations, grounded in the discriminatory property of with respect to and the separation of by . The results enable representations of shapes and other data via finite metric-space approximations (point clouds) by real-valued functions, providing a principled universal-approximation mechanism across variable domains.

Abstract

We establish sufficient conditions for the density of shallow neural networks \cite{C89} on the family of continuous real functions defined on a compact metric space, taking into account variations in the function domains. For this we use the Gromov-Hausdorff distance defined in \cite{5G}.

Paper Structure

This paper contains 3 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Proof of Theorem \ref{['pertinho']}

Key Result

Theorem 2

Let $\mathcal{F}$ a locally sliceable subset of $\mathcal{C}$ satisfying rosa and $\sigma$ be an activation function of sigmoidal type. If is dense in $D(\mathcal{F})$, then the shallow neural network generated by $\sigma$ and $\mathcal{F}$ is dense in $\mathcal{C}$.

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 2
  • Corollary 3
  • Example 1
  • Example 2
  • Example 3
  • ...and 22 more