Refined rates of convergence for target-data dependent greedy generalized interpolation with Sobolev kernels

TL;DR

This paper improves convergence rate estimates for target-data-dependent greedy kernel interpolation with Sobolev kernels by removing a logarithmic factor, enhancing efficiency especially in high-dimensional settings.

Contribution

It introduces a method to eliminate the logarithmic term in convergence rates using metric entropy estimates, refining previous bounds for Sobolev kernel interpolation.

Findings

Achieves dimension- and smoothness-independent convergence improvements

Removes the logarithmic factor from existing convergence rate bounds

Enhances the theoretical understanding of greedy kernel interpolation efficiency

Abstract

Greedy methods have recently been successfully applied to generalized kernel interpolation, or the recovery of a function from data stemming from the evaluation of linear functionals, including the approximation of solutions of linear PDEs by symmetric collocation. When applied to kernels generating Sobolev spaces as their native Hilbert spaces, some of these greedy methods can provide the same error guarantee of generalized interpolation on quasi-uniform points. More importantly, certain target-data-adaptive methods even give a dimension- and smoothness-independent improvement in the speed of convergence over quasi-uniform points, thus offering advantages for high-dimensional problems. These convergence rates however contain a spurious logarithmic term that limits this beneficial effect. The goal of this note is to remove this factor, and this is possible by using estimates on metric entropy numbers.