MEAN DIMENSION OF RADIAL BASIS FUNCTIONS

We show that generalized multiquadric radial basis functions (RBFs) on R d have a mean dimension that is 1 + O (1 /d ) as d -+ oo 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.