Piecewise Linear Functions Representable with Infinite Width Shallow ReLU Neural Networks

Abstract

This paper analyzes representations of continuous piecewise linear functions with infinite width, finite cost shallow neural networks using the rectified linear unit (ReLU) as an activation function. Through its integral representation, a shallow neural network can be identified by the corresponding signed, finite measure on an appropriate parameter space. We map these measures on the parameter space to measures on the projective $n$-sphere cross $\mathbb{R}$, allowing points in the parameter space to be bijectively mapped to hyperplanes in the domain of the function. We prove a conjecture of Ongie et al. that every continuous piecewise linear function expressible with this kind of infinite width neural network is expressible as a finite width shallow ReLU neural network.