Deep ReLU neural networks in high-dimensional approximation

We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the Hölder–Zygmund space of mixed smoothness defined on the d-dimensional unit cube when the dimension d may be very large. The approximation error is measured in the norm...

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Vydané v:	Neural networks Ročník 142; s. 619 - 635
Hlavní autori:	Dũng, Dinh, Nguyen, Van Kien
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier Ltd 01.10.2021
Predmet:	Computation complexity Continuous piece-wise linear functions Deep ReLU neural network High-dimensional approximation Hölder–Zygmund space of mixed smoothness Sparse-grid sampling High-dimensional approximation Computation complexity Deep ReLU neural network Hölder–Zygmund space of mixed smoothness Sparse-grid sampling Continuous piece-wise linear functions
ISSN:	0893-6080, 1879-2782, 1879-2782
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Popis
Shrnutí:	We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the Hölder–Zygmund space of mixed smoothness defined on the d-dimensional unit cube when the dimension d may be very large. The approximation error is measured in the norm of isotropic Sobolev space. For every function f from the Hölder–Zygmund space of mixed smoothness, we explicitly construct a deep ReLU neural network having an output that approximates f with a prescribed accuracy ɛ, and prove tight dimension-dependent upper and lower bounds of the computation complexity of the approximation, characterized as the size and depth of this deep ReLU neural network, explicitly in d and ɛ. The proof of these results in particular, relies on the approximation by sparse-grid sampling recovery based on the Faber series.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23
ISSN:	0893-6080 1879-2782 1879-2782
DOI:	10.1016/j.neunet.2021.07.027