Lp sampling numbers for the Fourier-analytic Barron space

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 → ∞.