mlsanityData to 7 Oct 2026

Representation formulas and pointwise properties for Barron functions

arXiv 2006.05982 ↗First posted 10 Jun 2020stat.ML cs.LG math.APcited by 62

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Abstract (arXiv)

We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only C1-diffeomorphisms which Barron space are affine. Furthermore, we show that every Barron function can be decomposed as the sum of a bounded and a positively one-homogeneous function and that there exist Barron functions which decay rapidly at infinity and are globally Lebesgue-integrable. This result suggests that two-layer neural networks may be able to approximate a greater variety of functions than commonly believed.

Key facts. Cited by 62 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Jun 2020.

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Citing papers per year: 2021: 11; 2022: 7; 2023: 13; 2024: 13; 2025: 5; 2026: 13.

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  1. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 95
  2. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  3. 2024Operator Learning: Algorithms and Analysiscited by 84
  4. 2021Characterization of the Variation Spaces Corresponding to Shallow Neural Networkscited by 48
  5. 2024Transformers Learn Nonlinear Features In Context: Nonconvex Mean-field Dynamics on the Attention Landscapecited by 43
  6. 2021Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neuronscited by 41
  7. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  8. 2021Two-layer neural networks with values in a Banach spacecited by 26
  9. 2023Sampling weights of deep neural networkscited by 20
  10. 2023Penalising the biases in norm regularisation enforces sparsitycited by 17
  11. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  12. 2022Duality for Neural Networks through Reproducing Kernel Banach Spacescited by 15
38 more
  1. 2022Small Transformers Compute Universal Metric Embeddingscited by 12
  2. 2024Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributionscited by 12
  3. 2022Understanding Deep Neural Function Approximation in Reinforcement Learning via ε-Greedy Explorationcited by 11
  4. 2021Optimal learning of high-dimensional classification problems using deep neural networkscited by 9
  5. 2024Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networkscited by 8
  6. 2023Infinite-dimensional reservoir computingcited by 7
  7. 2023ReLU Neural Networks with Linear Layers are Biased Towards Single- and Multi-Index Modelscited by 7
  8. 2024Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transportcited by 7
  9. 2022Lp sampling numbers for the Fourier-analytic Barron spacecited by 6
  10. 2022Universality and approximation bounds for echo state networks with random weightscited by 5
  11. 2023Neural Hilbert Ladders: Multi-Layer Neural Networks in Function Spacecited by 5
  12. 2023Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spacescited by 5
  13. 2024Stable Learning Using Spiking Neural Networks Equipped With Affine Encoders and Decoderscited by 5
  14. 2022Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionalitycited by 4
  15. 2023Embeddings between Barron spaces with higher order activation functionscited by 4
  16. 2021Tighter Sparse Approximation Bounds for ReLU Neural Networkscited by 3
  17. 2023Minimum norm interpolation by perceptra: Explicit regularization and implicit biascited by 3
  18. 2021Integral representations of shallow neural network with Rectified Power Unit activation functioncited by 2
  19. 2022Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activationcited by 2
  20. 2023Delay-SDE-net: A deep learning approach for time series modelling with memory and uncertainty estimatescited by 2
  21. 2023Embedding Inequalities for Barron-type Spacescited by 2
  22. 2024Approximation Bounds for Recurrent Neural Networks with Application to Regressioncited by 2
  23. 2024A Lipschitz spaces view of infinitely wide shallow neural networkscited by 2
  24. 2024High-dimensional classification problems with Barron regular boundaries under margin conditionscited by 2
  25. 2024Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspectivecited by 2
  26. 2025Nonlocal techniques for the analysis of deep ReLU neural network approximationscited by 2
  27. 2025Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operatorscited by 2
  28. 2021Sharp Lower Bounds on the Approximation Rate of Shallow Neural Networkscited by 1
  29. 2023Piecewise Linear Functions Representable with Infinite Width Shallow ReLU Neural Networkscited by 1
  30. 2023Barron Space for Graph Convolution Neural Networkscited by 1
  31. 2023Learning a Sparse Representation of Barron Functions with the Inverse Scale Space Flowcited by 1
  32. 2025Curse of Dimensionality in Neural Network Optimizationcited by 1
  33. 2025Scalable Complexity Control Facilitates Reasoning Ability of LLMscited by 1
  34. 2025Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditionscited by 1
  35. 2026Sobolev Approximation of Deep ReLU Networks in Log-Barron Spacecited by 1
  36. 2026Manifold limit for the training of shallow graph convolutional neural networkscited by 1
  37. 2026Regularity of Second-Order Elliptic PDEs in Spectral Barron Spacescited by 1
  38. 2026On the global convergence of gradient flow for wide shallow models beyond homogeneous nonlinearitiescited by 1

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