On the universal approximation property of radial basis function neural networks

TL;DR

This paper introduces a new class of radial basis function neural networks with shift-based parameters, proving their universal approximation capability for continuous functions on compact sets under certain conditions.

Contribution

It presents a novel RBF network model replacing smoothing factors with shifts and establishes conditions for universal approximation and arbitrary precision approximation with fixed centroids.

Findings

Networks can approximate any continuous multivariate function on compact sets.

Universal approximation is proven under specific conditions on the activation function.

Approximation with arbitrary precision is achievable with finitely many fixed centroids.

Abstract

In this paper we consider a new class of RBF (Radial Basis Function) neural networks, in which smoothing factors are replaced with shifts. We prove under certain conditions on the activation function that these networks are capable of approximating any continuous multivariate function on any compact subset of the $d$-dimensional Euclidean space. For RBF networks with finitely many fixed centroids we describe conditions guaranteeing approximation with arbitrary precision.