MEAN DIMENSION OF RADIAL BASIS FUNCTIONS.

We show that generalized multiquadric radial basis functions (RBFs) on ℝd have a mean dimension that is 1 + O(1/d) as d → ∞ 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. [ABSTRACT FROM AUTHOR]