# Explicit integral representations and quantitative bounds for two-layer ReLU networks

arXiv: https://arxiv.org/abs/2604.23260 · HTML: https://mlsanity.com/paper/2604.23260/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 19 references that are in the map.
- First posted on arXiv 25 Apr 2026.

## Builds on (most cited first, 10 of 10)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Neural tangent kernels, transportation mappings, and universal approximation](https://mlsanity.com/paper/1910.06956/) (2019)
- [On the Approximation Power of Two-Layer Networks of Random ReLUs](https://mlsanity.com/paper/2102.02336/) (2021)
- [Neural network integral representations with the ReLU activation function](https://mlsanity.com/paper/1910.02743/) (2019)

## Abstract (arXiv)

An approach to construct explicit integral representations for two-layer ReLU networks is presented, which provides relatively simple representations for any multivariate polynomial. Quantitative bounds are provided for a particular, sharpened ReLU integral representation, which involves a harmonic extension and a projection. The bounds demonstrate that functions can be approximated with L2(D) errors that do not depend explicitly on dimension or degree, but rather the coefficients of their monomial expansions and the distribution D. We also present a connection to the RKHS of the exponential kernel K(x,y)=expleft(leftlangle x,yrightrangle right), and a very simple integral representation involving additionally multiplication via a fixed function which has better quantitative bounds.