Mean dimension of radial basis functions

Abstract: 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.