# Neural network integral representations with the ReLU activation function

arXiv: https://arxiv.org/abs/1910.02743 · HTML: https://mlsanity.com/paper/1910.02743/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Oct 2019.

## Built on by (most cited first, 8 of 8)

- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Neural Control Variates with Automatic Integration](https://mlsanity.com/paper/2409.15394/) (2024)
- [Approximation with Random Shallow ReLU Networks with Applications to Model Reference Adaptive Control](https://mlsanity.com/paper/2403.17142/) (2024)
- [Integral representations of shallow neural network with Rectified Power Unit activation function](https://mlsanity.com/paper/2112.11157/) (2021)
- [Nonuniform random feature models using derivative information](https://mlsanity.com/paper/2410.02132/) (2024)
- [Towards Global Neural Network Abstractions with Locally-Exact Reconstruction](https://mlsanity.com/paper/2210.12054/) (2022)
- [Function Approximation with Randomly Initialized Neural Networks for Approximate Model Reference Adaptive Control](https://mlsanity.com/paper/2303.16251/) (2023)
- [Explicit integral representations and quantitative bounds for two-layer ReLU networks](https://mlsanity.com/paper/2604.23260/) (2026)

## Abstract (arXiv)

In this effort, we derive a formula for the integral representation of a shallow neural network with the ReLU activation function. We assume that the outer weighs admit a finite L1-norm with respect to Lebesgue measure on the sphere. For univariate target functions we further provide a closed-form formula for all possible representations. Additionally, in this case our formula allows one to explicitly solve the least L1-norm neural network representation for a given function.