Stability of convergence rates: Kernel interpolation on non-Lipschitz domains

TL;DR

This paper proves that kernel interpolation convergence rates are stable across arbitrary, including non-Lipschitz, domains, extending previous results and providing insights into approximation behavior for smooth kernels like Gaussian.

Contribution

It introduces a new analysis showing convergence rate stability of kernel interpolation on arbitrary domains, including irregular boundaries, using greedy kernel algorithms.

Findings

Convergence rates are stable when restricting to smaller domains.

Applicable to both finite and infinite smoothness kernels like Gaussian.

Numerical experiments confirm theoretical predictions.

Abstract

Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces (RKHS) usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we leverage an analysis of greedy kernel algorithms to prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $\Omega \subset \mathbb{R}^d$, thus allowing for non-Lipschitz domains including e.g. cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tilde{\Omega} \subset \Omega \subset \mathbb{R}^d$, the convergence rate does not deteriorate - i.e. the convergence rates are stable with respect to going to a subset. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness like the Gaussian kernel. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $\Omega$ has an impact on the approximation properties. Numerical experiments illustrate and confirm the experiments.