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

TL;DR

The paper addresses convergence-rate guarantees for kernel interpolation in Reproducing Kernel Hilbert Spaces on arbitrary domains, including non-Lipschitz geometries. By leveraging an abstract greedy-analysis framework and an extension operator that connects kernels on a large domain $\Omega$ to a subdomain $\tilde{\Omega}$, it proves that convergence rates in the number of interpolation points are preserved (or improved) under domain restriction. The main contribution is a general theorem: if interpolation enjoys algebraic or exponential decay on $\Omega$, then the same rate holds on $\tilde{\Omega}$ with controlled constants, applicable to kernels of finite and infinite smoothness such as Matérn and Gaussian. The results illuminate the robustness of RKHS-based kernel interpolation beyond classical Lipschitz-domain assumptions and are corroborated by numerical experiments on subdomains and non-Lipschitz domains. This advances the practical applicability of kernel methods in irregular geometries and offers a principled link between domain geometry and interpolation performance.

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 $Ω\subset \mathbb{R}^d$, thus allowing for non-Lipschitz domains including e.g. cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tildeΩ \subset Ω\subset \mathbb{R}^d$, 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 $Ω$ has an impact on the approximation properties. Numerical experiments illustrate and confirm the experiments.

Figures (3)

• Figure 1: Top: Visualization of the selected points for $P$-greedy applied to $\tilde{\Omega}$ (left) and $\Omega$ (right). Bottom: Visualization of the decay $\Vert P_{k, \Omega, X_n} \Vert_{L^\infty(\Omega)}$ and $\Vert P_{\tilde{k}, \tilde{\Omega}, X_n} \Vert_{L^\infty(\Omega)}$ ($y$-axis) in the number $n$ of selected interpolation points ($x$-axis). The convergence rate on the smaller domain $\tilde{\Omega}$ is faster compared to the larger domain $\Omega$.
• Figure 2: Visualization of the $P$-greedy algorithm applied to $\Omega$ and $\tilde{\Omega} \subset \Omega$ as defined in Eq. \ref{['eq:definition_domains']}: Left: Basic Matérn kernel, right: Linear Matérn kernel. On top the decay of $\Vert P_{k, \Omega, X_n} \Vert_{L^\infty(\Omega)}$ respectively $\Vert P_{\tilde{k}, \tilde{\Omega}, X_n} \Vert_{L^\infty(\tilde{\Omega})}$ is displayed. The $P$-greedy selected points are visualized in the middle row (for $\tilde{\Omega}$) and bottom row (for $\Omega$).
• Figure 3: Top: Visualization of the selected points for $P$-greedy applied to $\tilde{\Omega}$ (left) and $\Omega$ (right). Bottom: Visualization of the decay $\Vert P_{\tilde{k}, \tilde{\Omega}, X_n} \Vert_{L^\infty(\tilde{\Omega})}$ and $\Vert P_{k, \Omega, X_n} \Vert_{L^\infty(\Omega)}$ ($y$-axis) in the number $n$ of selected interpolation points ($x$-axis) for the Gaussian kernel. The convergence rate on the smaller domain $\tilde{\Omega}$ is clearly at least as fast as on the larger domain.

Theorems & Definitions (20)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• Theorem 4
• Theorem 5
• proof
• Corollary 6
• proof