Does the Barron space really defy the curse of dimensionality?

TL;DR

This paper investigates whether the Barron space truly overcomes the curse of dimensionality, revealing that its apparent advantages depend on a nonclassical smoothness concept linked to infinitely wide neural networks.

Contribution

The paper introduces ADZ spaces defined via the Mellin transform, providing evidence that Barron space's defiance of the curse depends on a nonclassical smoothness notion.

Findings

Barron space's curse-defying property is linked to nonclassical smoothness.

ADZ spaces encapsulate this nonclassical smoothness.

The results challenge the classical interpretation of Barron space's advantages.

Abstract

The Barron space has become famous in the theory of (shallow) neural networks because it seemingly defies the curse of dimensionality. And while the Barron space (and generalizations) indeed defies (defy) the curse of dimensionality from the POV of classical smoothness, we herein provide some evidence in favor of the idea that the Barron space (and generalizations) does (do) not defy the curse of dimensionality with a nonclassical notion of smoothness which relates naturally to "infinitely wide" shallow neural networks. Like how the Bessel potential spaces are defined via the Fourier transform, we define so-called ADZ spaces via the Mellin transform; these ADZ spaces encapsulate the nonclassical smoothness we alluded to earlier. 38 pages, will appear in the dissertation of the author