mlsanityData to 14 Oct 2026

The Barron Space and the Flow-induced Function Spaces for Neural Network Models

arXiv 1906.08039 ↗First posted 18 Jun 2019cs.LG math.PR stat.MLcited by 39

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Abstract (arXiv)

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: the two-layer networks and the residual neural networks. We define the Barron space and show that it is the right space for two-layer neural network models in the sense that optimal direct and inverse approximation theorems hold for functions in the Barron space. For residual neural network models, we construct the so-called flow-induced function space, and prove direct and inverse approximation theorems for this space. In addition, we show that the Rademacher complexity for bounded sets under these norms has the optimal upper bounds.

Key facts. Cited by 39 papers in the map (data through 14 Oct 2026). First posted on arXiv 18 Jun 2019.

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Citing papers per year: 2021: 14; 2022: 7; 2023: 5; 2024: 8; 2025: 3; 2026: 2.

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  1. 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
  2. 2021Characterization of the Variation Spaces Corresponding to Shallow Neural Networkscited by 48
  3. 2023What can a Single Attention Layer Learn? A Study Through the Random Features Lenscited by 40
  4. 2023Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Modelscited by 39
  5. 2021Two-layer neural networks with values in a Banach spacecited by 26
  6. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  7. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  8. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18
  9. 2024Wide Neural Networks Trained with Weight Decay Provably Exhibit Neural Collapsecited by 16
  10. 2022Duality for Neural Networks through Reproducing Kernel Banach Spacescited by 15
  11. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  12. 2021An L2 Analysis of Reinforcement Learning in High Dimensions with Kernel and Neural Network Approximationcited by 13
27 more
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  2. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  3. 2024Shallow diffusion networks provably learn hidden low-dimensional structurecited by 12
  4. 2024Random Projection Neural Networks of Best Approximation: Convergence theory and practical applicationscited by 11
  5. 2022Approximation of Functionals by Neural Network without Curse of Dimensionalitycited by 8
  6. 2021Universal Approximation Theorem for Neural Networkscited by 7
  7. 2022A Mathematical Framework for Learning Probability Distributionscited by 7
  8. 2022Importance Tempering: Group Robustness for Overparameterized Modelscited by 6
  9. 2024How DNNs break the Curse of Dimensionality: Compositionality and Symmetry Learningcited by 6
  10. 2025ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residualscited by 6
  11. 2022Bayesian Deep Learning with Multilevel Trace-class Neural Networkscited by 5
  12. 2025Deep Learning as a Convex Paradigm of Computation: Minimizing Circuit Size with ResNetscited by 5
  13. 2021Perturbational Complexity by Distribution Mismatch: A Systematic Analysis of Reinforcement Learning in Reproducing Kernel Hilbert Spacecited by 4
  14. 2021A Unified and Constructive Framework for the Universality of Neural Networkscited by 4
  15. 2023Reinforcement Learning with Function Approximation: From Linear to Nonlinearcited by 4
  16. 2023Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Normscited by 4
  17. 2025Saddle-To-Saddle Dynamics in Deep ReLU Networks: Low-Rank Bias in the First Saddle Escapecited by 4
  18. 2022The Deep Ritz Method for Parametric p-Dirichlet Problemscited by 3
  19. 2024Learning solution operators of PDEs defined on varying domains via MIONetcited by 3
  20. 2024A deformation-based framework for learning solution mappings of PDEs defined on varying domainscited by 2
  21. 2021Sharp Lower Bounds on the Approximation Rate of Shallow Neural Networkscited by 1
  22. 2021Sobolev-type embeddings for neural network approximation spacescited by 1
  23. 2023GIT-Net: Generalized Integral Transform for Operator Learningcited by 1
  24. 2022Optimal Learningcited by 0
  25. 2024Two-hidden-layer ReLU neural networks and finite elementscited by 0
  26. 2026Certified and accurate computation of function space norms of deep neural networkscited by 0
  27. 2026Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximationscited by 0