SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1191-1211, June 2024.
Abstract. We show that generalized multiquadric radial basis functions (RBFs) on [math] have a mean dimension that is [math] as [math] 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 [math]. 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 [math] increases.

中文翻译：

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径向基函数的平均维数


《SIAM 数值分析杂志》，第 62 卷，第 3 期，第 1191-1211 页，2024 年 6 月。

抽象的。我们证明，在输入的矩条件下，[math] 上的广义多二次径向基函数 (RBF) 的平均维度为 [math]，因为 [math] 具有隐含常数的显式界限。在较弱的矩条件下，平均维数仍然接近 1。因此，随着维数的增加，这些 RBF 本质上变得可加。相比之下，高斯 RBF 可以达到 1 到 [math] 之间的任何平均维度。我们还发现，Keister 的测试被积函数在准蒙特卡罗理论中具有影响力，其平均维数随着标称维数 [数学] 的增加而在大约 1 和大约 2 之间振荡。