# High-dimensional classification problems with Barron regular boundaries under margin conditions

arXiv: https://arxiv.org/abs/2412.07312 · HTML: https://mlsanity.com/paper/2412.07312/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 10 Dec 2024.

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

- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces](https://mlsanity.com/paper/2104.02746/) (2021)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Optimal learning of high-dimensional classification problems using deep neural networks](https://mlsanity.com/paper/2112.12555/) (2021)

## Built on by (most cited first, 2 of 2)

- [Minimax learning rates for estimating binary classifiers under margin conditions](https://mlsanity.com/paper/2505.10628/) (2025)
- [Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks](https://mlsanity.com/paper/2607.01266/) (2026)

## Abstract (arXiv)

We prove that a classifier with a Barron-regular decision boundary can be approximated with a rate of high polynomial degree by ReLU neural networks with three hidden layers when a margin condition is assumed. In particular, for strong margin conditions, high-dimensional discontinuous classifiers can be approximated with a rate that is typically only achievable when approximating a low-dimensional smooth function. We demonstrate how these expression rate bounds imply fast-rate learning bounds that are close to n-1 where n is the number of samples. In addition, we carry out comprehensive numerical experimentation on binary classification problems with various margins. We study three different dimensions, with the highest dimensional problem corresponding to images from the MNIST data set.