# Piecewise Linear Functions Representable with Infinite Width Shallow ReLU Neural Networks

arXiv: https://arxiv.org/abs/2307.14373 · HTML: https://mlsanity.com/paper/2307.14373/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 25 references that are in the map.
- First posted on arXiv 25 Jul 2023.

## Builds on (most cited first, 8 of 8)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Hidden Unit Specialization in Layered Neural Networks: ReLU vs. Sigmoidal Activation](https://mlsanity.com/paper/1910.07476/) (2019)

## Built on by (most cited first, 1 of 1)

- [Two-hidden-layer ReLU neural networks and finite elements](https://mlsanity.com/paper/2403.05809/) (2024)

## Abstract (arXiv)

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 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.