  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.
