mlsanityData to 7 Oct 2026

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

arXiv 2011.09363 ↗First posted 18 Nov 2020math.FA stat.MLcited by 28

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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.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). First posted on arXiv 18 Nov 2020.

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Citing papers per year: 2021: 7; 2022: 3; 2023: 5; 2024: 7; 2025: 4; 2026: 2.

28 papers.

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  1. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  2. 2021Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spacescited by 30
  3. 2024A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Modelscited by 20
  4. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  5. 2021Stationary Density Estimation of Ito Diffusions Using Deep Learningcited by 9
  6. 2021Optimal learning of high-dimensional classification problems using deep neural networkscited by 9
  7. 2022Learning Ability of Interpolating Deep Convolutional Neural Networkscited by 9
  8. 2022Lp sampling numbers for the Fourier-analytic Barron spacecited by 6
  9. 2023Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spacescited by 5
  10. 2024Stable Learning Using Spiking Neural Networks Equipped With Affine Encoders and Decoderscited by 5
  11. 2022Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionalitycited by 4
  12. 2023Embeddings between Barron spaces with higher order activation functionscited by 4
16 more
  1. 2023On Excess Risk Convergence Rates of Neural Network Classifierscited by 3
  2. 2023Minimum norm interpolation by perceptra: Explicit regularization and implicit biascited by 3
  3. 2021Integral representations of shallow neural network with Rectified Power Unit activation functioncited by 2
  4. 2023Embedding Inequalities for Barron-type Spacescited by 2
  5. 2024Do stable neural networks exist for classification problems? -- A new view on stability in AIcited by 2
  6. 2024High-dimensional classification problems with Barron regular boundaries under margin conditionscited by 2
  7. 2025Nonlocal techniques for the analysis of deep ReLU neural network approximationscited by 2
  8. 2025Time-Frequency Analysis for Neural Networkscited by 2
  9. 2021Sobolev-type embeddings for neural network approximation spacescited by 1
  10. 2024Universality of reservoir systems with recurrent neural networkscited by 1
  11. 2025Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equationscited by 1
  12. 2026Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Methodcited by 1
  13. 2024Dimension-independent learning rates for high-dimensional classification problemscited by 0
  14. 2024ReLU neural network approximation to piecewise constant functionscited by 0
  15. 2025Minimax learning rates for estimating binary classifiers under margin conditionscited by 0
  16. 2026Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networkscited by 0