# Lp sampling numbers for the Fourier-analytic Barron space

arXiv: https://arxiv.org/abs/2208.07605 · HTML: https://mlsanity.com/paper/2208.07605/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 16 Aug 2022.

## Builds on (most cited first, 7 of 7)

- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces](https://mlsanity.com/paper/2104.02746/) (2021)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networks](https://mlsanity.com/paper/1607.01434/) (2016)
- [Optimal learning of high-dimensional classification problems using deep neural networks](https://mlsanity.com/paper/2112.12555/) (2021)

## Built on by (most cited first, 6 of 6)

- [Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces](https://mlsanity.com/paper/2312.08461/) (2023)
- [Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains](https://mlsanity.com/paper/2411.04108/) (2024)
- [Nonlocal techniques for the analysis of deep ReLU neural network approximations](https://mlsanity.com/paper/2504.04847/) (2025)
- [Time-Frequency Analysis for Neural Networks](https://mlsanity.com/paper/2512.15992/) (2025)
- [Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method](https://mlsanity.com/paper/2607.25100/) (2026)
- [Approximation Rates for Metaplectic Neural Networks](https://mlsanity.com/paper/2608.08872/) (2026)

## Abstract (arXiv)

In this paper, we consider Barron functions f : [0,1]d → ℝ of smoothness σ > 0, which are functions that can be written as [ f(x) = intℝd F(xi) , e2 π i langle x, xi rangle , d xi quad with quad intℝd |F(xi)| · (1 + |xi|)σ , d xi < ∞. ] For σ = 1, these functions play a prominent role in machine learning, since they can be efficiently approximated by (shallow) neural networks without suffering from the curse of dimensionality. For these functions, we study the following question: Given m point samples f(x1),dots,f(xm) of an unknown Barron function f : [0,1]d → ℝ of smoothness σ, how well can f be recovered from these samples, for an optimal choice of the sampling points and the reconstruction procedure? Denoting the optimal reconstruction error measured in Lp by sm (σ; Lp), we show that [ m^- frac1max  p,2  - fracσd lesssim sm(σ;Lp) lesssim (ln (e + m))α(σ,d) / p · m^- frac1max  p,2  - fracσd , ] where the implied constants only depend on σ and d and where α(σ,d) stays bounded as d → ∞.