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This paper analyzes representations of continuous piecewise linear functionswith infinite width, finite cost shallow neural networks using the rectifiedlinear unit (ReLU) as an activation function. Through its integralrepresentation, a shallow neural network can be identified by the correspondingsigned, finite measure on an appropriate parameter space. We map these measureson the parameter space to measures on the projective $n$-sphere cross$\mathbb{R}$, allowing points in the parameter space to be bijectively mappedto hyperplanes in the domain of the function. We prove a conjecture of Ongie etal. that every continuous piecewise linear function expressible with this kindof infinite width neural network is expressible as a finite width shallow ReLUneural network.

Piecewise Linear Functions Representable with Infinite Width Shallow ReLU Neural Networks