Mean dimension of radial basis functions
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We show that generalized multiquadric radial basis functions (RBFs) on ℝ^d 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.
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