论文标题
$ l^p $采样数字，用于傅立叶分析巴伦空间
$L^p$ sampling numbers for the Fourier-analytic Barron space
论文作者
论文摘要
在本文中，我们考虑Barron函数$ f：[0,1]^d \ to \ mathbb {r} $ of平滑度$σ> 0 $，这是可以写为\ [ f（x）= \ int _ {\ mathbb {r}^d} f（ξ）\，e^{2πi\ langle x，ξ\ rangle} \，d en \ quad \ text {with} \ quad \ int _ {\ mathbb {r}^d} | f（ξ）| \ cdot（1 + |ξ|）^σ\，dξ<\ infty。 \]对于$σ= 1 $，这些功能在机器学习中起着重要的作用，因为它们可以通过（浅）神经网络有效地近似，而不会受到维度的诅咒。 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?表示在$ l^p $ by $ s_m（σ; l^p）$中测量的最佳重建错误，我们显示\ [ m^{ - \ frac {1} {\ max \ {p，2 \}} - \fracσ{d}}} \ Lessim s_m（σ; l^p） \ Lessim（\ ln（e + m））^{α（σ，d） / p} \ cdot m^{ - \ frac {1} {\ max \ {p，2 \}} - \fracσ{d}}} ，\]中暗示常数仅取决于$σ$和$ d $，而$α（σ，d）$保持为$ d \ to \ infty $。
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 $d \to \infty$.
