We show that generalized multiquadric radial basis functions (RBFs) on $\mathbb{R}^d$ have a mean dimension that is $1+O(1/d)$ as $d\to\infty$ with an explicit bound for the implied constant, under moment conditions on their inputs. Under weaker moment conditions the mean dimension still approaches $1$. As a consequence, these RBFs become essentially additive as their dimension increases. Gaussian RBFs by contrast can attain any mean dimension between 1 and d. We also find that a test integrand due to Keister that has been influential in quasi-Monte Carlo theory has a mean dimension that oscillates between approximately 1 and approximately 2 as the nominal dimension $d$ increases.

翻译：我们显示,对$mathbb{R ⁇ d$的普遍多赤道半径基功能(RBFs)的平均值为$+O(1/d)$($+O(1d)$),在输入的瞬间条件下,以隐含的恒定值为明确约束。在较弱的时段,平均维度仍然接近$1。因此,这些RBFs随着其维度的增加而基本变异。对比之下,这些RBFs可以达到1美元到d之间的任何中位维度。 我们还发现,在准蒙特卡洛理论中具有影响力的Keister的测试中,有一个平均维度,其名义维度在大约1美元到2美元之间上升。