Characterization of the Variation Spaces Corresponding to Shallow Neural Networks

We study the variation space corresponding to a dictionary of functions in L 2 ( Ω ) for a bounded domain Ω ⊂ R d . Specifically, we compare the variation space, which is defined in terms of a convex hull with related notions based on integral representations. This allows us to show that three impor...

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Bibliographic Details
Published inConstructive approximation Vol. 57; no. 3; pp. 1109 - 1132
Main Authors Siegel, Jonathan W., Xu, Jinchao
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2023
Springer Nature B.V
Subjects
Analysis
Convexity
Dictionaries
Mathematics
Mathematics and Statistics
Neural networks
Numerical Analysis
Approximation
Function space
68T05
Neural networks
46B99
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Summary:We study the variation space corresponding to a dictionary of functions in L 2 ( Ω ) for a bounded domain Ω ⊂ R d . Specifically, we compare the variation space, which is defined in terms of a convex hull with related notions based on integral representations. This allows us to show that three important notions relating to the approximation theory of shallow neural networks, the Barron space, the spectral Barron space, and the Radon BV space, are actually variation spaces with respect to certain natural dictionaries.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-023-09626-4