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  4. Approximation of Smoothness Classes by Deep Rectifier Networks
 
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2021
Journal Article
Titel

Approximation of Smoothness Classes by Deep Rectifier Networks

Abstract
We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $Balpha_{q}(Lp)$ in arbitrary dimension $d$, on general domains. We show that deep rectifier networks with a fixed activation function attain optimal or near to optimal approximation rates for functions in the Besov space $Balpha_{\tau}(Ltau)$ on the critical embedding line $1/\tau=\alpha/d+1/p$ for arbitrary smoothness order $\alpha>0$. Using interpolation theory, this implies that the entire range of smoothness classes at or above the critical line is (near to) optimally approximated by deep ReLU/RePU networks.
Author(s)
Ali, Mazen
Fraunhofer-Institut fĂĽr Techno- und Wirtschaftsmathematik ITWM
Nouy, Anthony
Centrale Nantes, Department of Mathematics and Computer Science Laboratoire de Mathématiques,Jean Leray
Zeitschrift
SIAM journal on numerical analysis
Thumbnail Image
DOI
10.1137/20M1360657
Language
English
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Fraunhofer-Institut fĂĽr Techno- und Wirtschaftsmathematik ITWM
Tags
  • ReLU neural networks

  • approximation spaces

  • Besov spaces

  • direct embeddings

  • direct (Jackson) inequalities

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