Characterization of the Variation Spaces Corresponding to Shallow Neural Networks

Jonathan W. Siegel*, Jinchao Xu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1109-1132
Number of pages24
JournalConstructive Approximation
Issue number3
StatePublished - Jun 2023


  • Approximation
  • Function space
  • Neural networks

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics


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