Rate of Convergence and Tractability of the Radial Function Approximation Problem

This article studies the problem of approximating functions belonging to a Hilbert space Hd with an isotropic or anisotropic Gaussian reproducing kernel, Kd(x, t) = exp ( − d ∑ l=1 γ l (xl − tl)2 ) for all x, t ∈ R. The isotropic case corresponds to using the same shape parameters for all coordinates, namely γl = γ > 0 for all l, whereas the anisotropic case corresponds to varying shape parameters γl. We are especially interested in moderate to large d. We consider two classes of algorithms: (1) using finitely many arbitrary linear functionals, (2) using only finitely many function values. The pertinent error criterion is the worst case of such an algorithm over the unit ball in Hd, with the error for a single function given by the L2 norm also with a Gaussian weight. Since the Gaussian kernel is analytic, the minimal worst case errors of algorithms that use at most n linear functionals or n function values vanish like O(n−p) as n goes to infinity. Here, p can be arbitrarily large, but the leading coefficient may depend on d (Theorem 1). On the other hand, if d dependence is taken into account, the convergence rate may be quite slow. If the goal is to make the error smaller than Cn for some C independent of d or polynomially dependent on d, then this is possible for any choice of shape parameters with the largest p equal to 1/2, provided that arbitrary linear functional data is available (Theorem 2). If the sequence of shape parameters γl decays to zero like l −ω as l (and therefore also d) tends to ∞, then the largest p is roughly max(1/2, ω) (Theorem 3). If only function values are available, dimension-independent convergence rates are somewhat worse (Theorems 4 and 5). If the goal is to make the error smaller than Cn times the initial (n = 0) error, then the corresponding p is roughly ω. Therefore it is the same as before iff ω ≥ 1/2 (Theorem 7 and Corollary 2). In particular, for the isotropic case, when ω = 0, the error does not even decay polynomially with n (Theorem 6). In summary, excellent dimension independent error decay rates are only possible when the sequence of shape parameters decays rapidly. Date: May, 2010. 2010 Mathematics Subject Classification. Primary 65D15; Secondary 68Q17, 41A25, 41A63.