mlsanityData to 14 Oct 2026

Shallow ReLUs Networks in Lp-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization

arXiv 2605.18468 ↗First posted 18 May 2026stat.ML cs.LGcited by 2

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Abstract (arXiv)

This paper studies approximation by shallow ReLUs networks, σs(t)=max0,ts, together with their generalization behavior under ℓ1 path-norm control. For the Lp-type integral spaces widetildeFp,τd,s, 1≤ p≤2, spherical harmonic analysis yields approximation bounds for shallow networks. In particular, when τd is the uniform measure and 1≤ p<2, the approximation rate is O!left(m^-fracp(2s+2d+1)-2d2dpright) for 1≤ p≤ p^* and O!left(m^-fracp(4s+3d-1)-2d+24dpright) for p^*<p<2, where p^*=frac2d+2d+3. Approximation bounds for Sobolev spaces Wα,p, 1≤ p<2, are obtained through embeddings into spectral Barron spaces. For nonparametric regression with sub-Gaussian noise, path-norm-regularized shallow ReLUs networks achieve minimax-optimal rates O!left(n^-fracd+2s+12d+2s+1log nright) over mathscrBs and O!left(n^-frac2α2α+dlog nright) over Wα,∞, with matching lower bounds up to logarithmic factors.

Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 19 of its 44 references that are in the map. First posted on arXiv 18 May 2026.

Builds on

19 of 44 references.

References · 19

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7 more
  1. 2020Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Datacited by 22
  2. 2023Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networkscited by 11
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networkscited by 0
  2. 2026Minimax rates for learning spectral Barron functions by deep ReLU neural networkscited by 0