# ReLU neural network approximation to piecewise constant functions

arXiv: https://arxiv.org/abs/2410.16506 · HTML: https://mlsanity.com/paper/2410.16506/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 13 references that are in the map.
- First posted on arXiv 21 Oct 2024.

## Builds on (most cited first, 6 of 6)

- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Deep Neural Networks Learn Non-Smooth Functions Effectively](https://mlsanity.com/paper/1802.04474/) (2018)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)
- [Least-Squares Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Laws: Discrete Divergence Operator](https://mlsanity.com/paper/2110.10895/) (2021)

## Abstract (arXiv)

This paper studies the approximation property of ReLU neural networks (NNs) to piecewise constant functions with unknown interfaces in bounded regions in ℝd. Under the assumption that the discontinuity interface Gamma may be approximated by a connected series of hyperplanes with a prescribed accuracy varepsilon >0, we show that a three-layer ReLU NN is sufficient to accurately approximate any piecewise constant function and establish its error bound. Moreover, if the discontinuity interface is convex, an analytical formula of the ReLU NN approximation with exact weights and biases is provided.