Mean dimension of radial basis functions

TL;DR

This paper analyzes the mean dimension of radial basis functions, showing that generalized multiquadric RBFs become essentially additive in high dimensions, while Gaussian RBFs can have a wide range of mean dimensions.

Contribution

It provides explicit bounds and asymptotic behavior of the mean dimension for multiquadric and Gaussian RBFs as the dimension grows.

Findings

Multiquadric RBFs have mean dimension approaching 1 as dimension increases.

Gaussian RBFs can have mean dimension between 1 and d, depending on parameters.

Keister test integrand's mean dimension oscillates between 1 and 2 with increasing dimension.

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.