Characterization of the Variation Spaces Corresponding to Shallow Neural Networks

Jonathan W. Siegel; Jinchao Xu

doi:10.1007/s00365-023-09626-4

Characterization of the Variation Spaces Corresponding to Shallow Neural Networks

Jonathan W. Siegel
, Jinchao Xu

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

We study the variation space corresponding to a dictionary of functions in L²(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.

Original language	English (US)
Pages (from-to)	1109-1132
Number of pages	24
Journal	Constructive Approximation
Volume	57
Issue number	3
DOIs	https://doi.org/10.1007/s00365-023-09626-4
State	Published - Jun 2023

All Science Journal Classification (ASJC) codes

Analysis
General Mathematics
Computational Mathematics

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10.1007/s00365-023-09626-4

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abstract = "We study the variation space corresponding to a dictionary of functions in L2(Ω) for a bounded domain Ω ⊂ Rd. 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.",

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Characterization of the Variation Spaces Corresponding to Shallow Neural Networks

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