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Deep ReLU network approximation of functions on a manifold

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arxiv 1908.00695 v1 pith:ZPI6IUKR submitted 2019-08-02 stat.ML cs.LG

Johannes Schmidt-Hieber This is my paper · ORCID
classification stat.MLcs.LG
keywords epsilonmanifoldnetworkapproximationbetadeepfunctionsparameters
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abstract

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 $\beta$ up to error $\epsilon$ using of the order of $\epsilon^{-d^*/\beta}\log(1/\epsilon)$ 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.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Geometry of Neural Reinforcement Learning in Continuous State and Action Spaces

    cs.LG 2025-07 conditional novelty 7.0 of 10

    For wide two-layer linearized neural policies in deterministic continuous RL, the locally attainable states concentrate on a manifold of dimension at most 2da+1, independent of the state dimension.

  2. Phase Transition in Nonparametric Minimax Rates for Covariate Shifts on Approximate Manifolds

    math.ST 2025-07 conditional novelty 7.0 of 10

    Under covariate shift with target data near a smooth d-dimensional manifold in D dimensions, the minimax regression rate switches between a manifold-dominated and a noise-dominated regime at a threshold set by source ...

  3. Dimension independent bounds for general shallow networks

    cs.LG 2019-08 conditional novelty 7.0 of 10

    An abstract theorem gives dimension-independent, smoothness-improving approximation rates for shallow kernel networks, covering ReLU networks, RBFs, manifold learning, and quasirandom integration.

  4. Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds

    math.NA 2025-05 reject novelty 6.0 of 10

    Proves an n^{-1/(d+2)}-type convergence rate for weak PINNs approximating entropy solutions of geometry-compatible conservation laws on d-dimensional manifolds, with network complexity independent of the ambient dimension.

  5. Dimension-independent rates for structured neural density estimation

    stat.ML 2024-11 conditional novelty 6.0 of 10

    Neural density estimators that factor over a known Markov random field achieve dimension-independent L1 rates n^{-1/(4+r)} (and optimally n^{-1/(2+r)}), where r is the maximum clique size.

  6. Estimation of a function of low local dimensionality by deep neural networks

    stat.ML 2019-08 accept novelty 6.0 of 10

    For regression functions that are locally low-dimensional, sparse neural network estimates achieve the rate n^{-2p/(2p+d*)}, with the exponent depending only on the local dimension d* and not the input dimension d.

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