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

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

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