Data to 7 Oct 2026

Deep ReLU network approximation of functions on a manifold

arXiv 1908.00695 ↗First posted 2 Aug 2019stat.ML cs.LGcited by 73

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Abstract (arXiv)

Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a d^*-dimensional manifold that is embedded into a space with potentially much larger ambient dimension. It is shown that sparsely connected deep ReLU networks can approximate a H"older function with smoothness index β up to error ε using of the order of ε-d^*/βlog(1/ε) many non-zero network parameters. As an application, we derive statistical convergence rates for the estimator minimizing the empirical risk over all possible choices of bounded network parameters.

Key facts. Cited by 73 papers in the map (data through 7 Oct 2026). First posted on arXiv 2 Aug 2019.

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Citing papers per year: 2021: 9; 2022: 12; 2023: 18; 2024: 19; 2025: 7; 2026: 8.

73 papers.

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2023Diffusion Models are Minimax Optimal Distribution Estimatorscited by 159
2022A Review of Off-Policy Evaluation in Reinforcement Learningcited by 92
2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
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2024Flow matching achieves almost minimax optimal convergencecited by 27
2023Deep Operator Learning Lessens the Curse of Dimensionality for PDEscited by 18
2021On the rate of convergence of a classifier based on a Transformer encodercited by 16
2025Understanding In-Context Learning on Structured Manifolds: Bridging Attention to Kernel Methodscited by 13
2021A likelihood approach to nonparametric estimation of a singular distribution using deep generative modelscited by 12
2022Effects of Data Geometry in Early Deep Learningcited by 12
2022Adaptive deep learning for nonlinear time series modelscited by 11
2023Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustnesscited by 10

38 more

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2021Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on H"older Classcited by 7
2024Posterior concentrations of fully-connected Bayesian neural networks with general priors on the weightscited by 7
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2023Sup-Norm Convergence of Deep Neural Network Estimator for Nonparametric Regression by Adversarial Trainingcited by 6
2024Regularized DeepIV with Model Selectioncited by 6
2024How DNNs break the Curse of Dimensionality: Compositionality and Symmetry Learningcited by 6
2021Estimation of a regression function on a manifold by fully connected deep neural networkscited by 5
2021Efficient Estimation in NPIV Models: A Comparison of Various Neural Networks-Based Estimatorscited by 5
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2024Dimension-independent rates for structured neural density estimationcited by 5
2021Non-asymptotic Excess Risk Bounds for Classification with Deep Convolutional Neural Networkscited by 4
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2023Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functionscited by 4
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2024Deep Transfer Learning: Model Framework and Error Analysiscited by 4
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2024Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimationcited by 3
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2024Breaking the curse of dimensionality in structured density estimationcited by 3
2025Transformers for Learning on Noisy and Task-Level Manifolds: Approximation and Generalization Insightscited by 3
2025Adjustment for Confounding using Pre-Trained Representationscited by 3
2025Geometry of Neural Reinforcement Learning in Continuous State and Action Spacescited by 3
2025Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networkscited by 3
2021A manifold learning approach for gesture recognition from micro-Doppler radar measurementscited by 2
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