# Neural network approximation and estimation of classifiers with classification boundary in a Barron class

arXiv: https://arxiv.org/abs/2011.09363 · HTML: https://mlsanity.com/paper/2011.09363/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2020.

## Built on by (most cited first, 28 of 28)

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- [Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method](https://mlsanity.com/paper/2607.25100/) (2026)
- [Dimension-independent learning rates for high-dimensional classification problems](https://mlsanity.com/paper/2409.17991/) (2024)
- [ReLU neural network approximation to piecewise constant functions](https://mlsanity.com/paper/2410.16506/) (2024)
- [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 bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization using networks of a suitable size, depending on the number of training samples available. The obtained approximation and estimation rates are independent of the dimension of the input, showing that the curse of dimensionality can be overcome in this setting; in fact, the input dimension only enters in the form of a polynomial factor. Regarding the regularity of the target classification function, we assume the interfaces between the different classes to be locally of Barron-type. We complement our results by studying the relations between various Barron-type spaces that have been proposed in the literature. These spaces differ substantially more from each other than the current literature suggests.