Rate of Convergence and Tractability of the Radial Function Approximation Problem
Provides theoretical convergence and tractability results for radial function approximation, relevant to practitioners choosing kernel parameters in high-dimensional settings.
The paper derives convergence rates for approximating functions in Hilbert spaces with isotropic and anisotropic Gaussian kernels, showing that the isotropic case suffers from the curse of dimensionality while the anisotropic case can be tractable for moderate to large d.
This article studies the problem of approximating functions belonging to a Hilbert space $H_d$ with an isotropic or anisotropic Gaussian reproducing kernel, $$ K_d(\bx,\bt) = \exp\left(-\sum_{\ell=1}^dγ_\ell^2(x_\ell-t_\ell)^2\right) \ \ \ \mbox{for all}\ \ \bx,\bt\in\reals^d. $$ The isotropic case corresponds to using the same shape parameters for all coordinates, namely $γ_\ell=γ>0$ for all $\ell$, whereas the anisotropic case corresponds to varying shape parameters $γ_\ell$. We are especially interested in moderate to large $d$.