# Dimension-independent learning rates for high-dimensional classification problems

arXiv: https://arxiv.org/abs/2409.17991 · HTML: https://mlsanity.com/paper/2409.17991/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 39 references that are in the map.
- First posted on arXiv 26 Sep 2024.

## Builds on (most cited first, 17 of 17)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [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)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory](https://mlsanity.com/paper/2105.03361/) (2021)
- [Deep Neural Networks Learn Non-Smooth Functions Effectively](https://mlsanity.com/paper/1802.04474/) (2018)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients](https://mlsanity.com/paper/1809.07321/) (2018)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Advantage of Deep Neural Networks for Estimating Functions with Singularity on Hypersurfaces](https://mlsanity.com/paper/2011.02256/) (2020)
- [Optimal learning of high-dimensional classification problems using deep neural networks](https://mlsanity.com/paper/2112.12555/) (2021)
- [A priori estimates for classification problems using neural networks](https://mlsanity.com/paper/2009.13500/) (2020)
- [Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems](https://mlsanity.com/paper/2206.00934/) (2022)

## Abstract (arXiv)

We study the problem of approximating and estimating classification functions that have their decision boundary in the RBV2 space. Functions of RBV2 type arise naturally as solutions of regularized neural network learning problems and neural networks can approximate these functions without the curse of dimensionality. We modify existing results to show that every RBV2 function can be approximated by a neural network with bounded weights. Thereafter, we prove the existence of a neural network with bounded weights approximating a classification function. And we leverage these bounds to quantify the estimation rates. Finally, we present a numerical study that analyzes the effect of different regularity conditions on the decision boundaries.