On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces
Armin Iske
TL;DR
Problem: establish convergence of kernel regression in a reproducing kernel Hilbert space from irregular samples on a compact domain. Approach: prove convergence in the RKHS norm under minimalistic conditions on the kernel $K$, the target $f\in\mathcal{H}_{K,\Omega}$, and the sample sequence, and derive uniform convergence rates under local Hölder continuity via the fill distance $h_n$. Findings: $\|s_n-f\|_K \to 0$ as $h_n \to 0$ for all $f\in\mathcal{H}_{K,\Omega}$, and $\|s_n-f\|_\infty = O(h_n^{\alpha/2})$ when $K$ is locally $\alpha$-Holder; rates depend on the Hölder exponent and the continuity of $K$. Significance: demonstrates convergence of kernel-based interpolation under very weak conditions, clarifies the role of sampling density via the fill distance, and discusses theoretical boundaries such as the non-embedding of all continuous functions into RKHS. Future work: explore even weaker conditions on $K$ and $X$, and investigate when a given $f\in\mathscr{C}(\Omega)$ lies in $\mathcal{H}_{K,\Omega}$ and what this implies for the kernel.
Abstract
We analyse the convergence of sampling algorithms for functions in reproducing kernel Hilbert spaces (RKHS). To this end, we discuss approximation properties of kernel regression under minimalistic assumptions on both the kernel and the input data. We first prove error estimates in the kernel's RKHS norm. This leads us to new results concerning uniform convergence of kernel regression on compact domains. For Lipschitz continuous and Hölder continuous kernels, we prove convergence rates.