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Sobolev-type embeddings for neural network approximation spaces

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Grohs, Philipp ; Voigtlaender, Felix:
Sobolev-type embeddings for neural network approximation spaces.
2021. - 14 S.

Volltext

Open Access
Volltext Link zum Volltext (externe URL):
https://arxiv.org/abs/2110.15304

Kurzfassung/Abstract

We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in Lp) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers. We prove embedding theorems between these spaces for different values of p. Furthermore, we derive sharp embeddings of these approximation spaces into Hölder spaces. We find that, analogous to the case of classical function spaces (such as Sobolev spaces, or Besov spaces) it is possible to trade "smoothness" (i.e., approximation rate) for increased integrability.
Combined with our earlier results in [arXiv:2104.02746], our embedding theorems imply a somewhat surprising fact related to "learning" functions from a given neural network space based on point samples: if accuracy is measured with respect to the uniform norm, then an optimal "learning" algorithm for reconstructing functions that are well approximable by ReLU neural networks is simply given by piecewise constant interpolation on a tensor product grid.

Weitere Angaben

Publikationsform:Preprint, Working paper, Diskussionspapier
Sprache des Eintrags:Englisch
Institutionen der Universität:Mathematisch-Geographische Fakultät > Mathematik > Lehrstuhl für Reliable Machine Learning
DOI / URN / ID:arXiv:2110.15304
Open Access: Freie Zugänglichkeit des Volltexts?:Ja
Titel an der KU entstanden:Nein
KU.edoc-ID:29923
Eingestellt am: 30. Mär 2022 14:24
Letzte Änderung: 31. Mär 2022 13:14
URL zu dieser Anzeige: https://edoc.ku.de/id/eprint/29923/
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