mlsanityData to 14 Oct 2026

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

arXiv 2412.07312 ↗First posted 10 Dec 2024cs.LG math.PR stat.MLcited by 2

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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.

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

8 of 20 references.

References · 8

  1. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 179
  2. 2020Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don'tcited by 95
  3. 2018A Priori Estimates of the Population Risk for Two-layer Neural Networkscited by 76
  4. 2020Representation formulas and pointwise properties for Barron functionscited by 62
  5. 2018Fast convergence rates of deep neural networks for classificationcited by 46
  6. 2021Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spacescited by 30
  7. 2020Neural network approximation and estimation of classifiers with classification boundary in a Barron classcited by 28
  8. 2021Optimal learning of high-dimensional classification problems using deep neural networkscited by 9

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 1.

2 papers.

  1. 2025Minimax learning rates for estimating binary classifiers under margin conditionscited by 0
  2. 2026Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networkscited by 0