$L^p$ sampling numbers for the Fourier-analytic Barron space

In this paper, we consider Barron functions $f : [0,1]^d \to \mathbb{R}$ of smoothness $\sigma > 0$, which are functions that can be written as $$f(x) = \int_{\mathbb{R}^d} F(\xi) \, e^{2 \pi i \langle x, \xi \rangle} \, d \xi \quad \text{with} \quad \int_{\mathbb{R}^d} |F(\xi)| \cdot (1 + |\xi|)^{\sigma} \, d \xi < \infty.$$ For $\sigma = 1$, 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 $f(x_1),\dots,f(x_m)$ of an unknown Barron function $f : [0,1]^d \to \mathbb{R}$ of smoothness $\sigma$, 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 $L^p$ by $s_m (\sigma; L^p)$, we show that $$m^{- \frac{1}{\max \{ p,2 \}} - \frac{\sigma}{d}} \lesssim s_m(\sigma;L^p) \lesssim (\ln (e + m))^{\alpha(\sigma,d) / p} \cdot m^{- \frac{1}{\max \{ p,2 \}} - \frac{\sigma}{d}} ,$$ where the implied constants only depend on $\sigma$ and $d$ and where $\alpha(\sigma,d)$ stays bounded as $d \to \infty$.