Sharp inverse statements for kernel interpolation

TL;DR

The paper addresses inverse statements in kernel-based interpolation on compact Lipschitz domains, establishing a sharp, one-to-one link between the smoothness of a function and its interpolation rate when using finitely smooth kernels whose RKHS is norm-equivalent to a Sobolev space \(H^\\tau(\\Omega)\\). By developing utility results that transfer continuous \(L^2(\\Omega)\\) error rates to discrete \(L^2(X)\\) rates via well-distributed point sets, it proves that an \(L^2\) interpolation rate of order \\(h_{X,\\Omega}^\\beta\\) implies Sobolev regularity \(f \\\in H^{\\beta'}(\\Omega)\\) for all \(\\beta' \\in (0,\\beta)\\; and if \(\\beta > \\tau\\), then \(f \\in \\mathcal{H}_k(\\Omega) \\simeq H^\\tau(\\Omega)\\). This yields a sharp, data-dependent description of smoothness from observed convergence, filling gaps left by prior nonsharp results on Lipschitz domains and extending sphere-based sharp results to general domains. A Matérn-type numerical example demonstrates the predicted decay rates and the one-to-one correspondence between smoothness and approximation rate. The approach opens avenues for kernel quadrature, PDE approximation with kernels, and statistical learning contexts, with further work on analytic kernels and adaptive sampling schemes.

Abstract

While direct statements for kernel based interpolation on regions $Ω\subset \mathbb{R}^d$ are well researched, far less is known about corresponding inverse statements. The available inverse statements for kernel based interpolation so far are not sharp. In this paper, we derive sharp inverse statements for interpolation using finitely smooth kernels, such as popular radial basis function (RBF) kernels like the class of Matérn or Wendland kernels. In particular, the results show that there is a one-to-one correspondence between the smoothness of a function and its approximation rate via kernel interpolation: If a function can be approximated with a given rate, it has a corresponding smoothness and vice versa.

Figures (2)

• Figure 1: Visualization of the $L^2(\Omega)$ error decay in the ($y$-axis) for interpolation with the Matérn kernel $k(x, z) = \exp(-\Vert x - z \Vert)$ on $\Omega = [0,1]$ in the number of equidistant interpolation points ($x$-axis) for different $\sigma$ parameters of the functions $x^\sigma \cdot (1-x)^\sigma$. This class of functions scales linearly in the Sobolev spaces $H^\tau(\Omega)$: It holds $x^\sigma \cdot (1-x)^\sigma \in H^{\sigma + 1/2}(\Omega)$$\forall \sigma' \in (0, \sigma)$. The dashed black lines indicate the convergence rates $n^{-0.71}, n^{-0.86}, n^{-1.01}$, $n^{-1.31}$, $n^{-1.61}$ and $n^{-2.01}$, while the legend lists numerically computed convergence rates.
• Figure 2: Visualization of the key steps of the proof of \ref{['th:inverse_statement_generalized']}. For the sake of a helpful visualization, $q=0.2$ was chosen, despite it does not match the assumptions. Top left: Initial set $Y_N$ (gray), reference set $Y_0$ (red) with circles of radius $\frac{1}{2} q = 0.1$ and set $Y_1'$ (purple, see Eq. \ref{["eq:definition_y'"]}). Top right: Set $Y_1"$ (purple, see Eq. \ref{['eq:definition_y"']}) and clusters of points refering to the sets $T_i^{(N)}$ (gray, see Eq. \ref{['eq:proof:definition_TiN']}). Bottom left: Restriction of the sets $T_i^{(N)}$ to $\tilde{T}_i^{(N)}$, see Eq. \ref{['eq:proof:definition_TiN_tilde']}. Bottom right: Additional visualization of $Y_1$ (red crosses, see Eq. \ref{['eq:set_Y1']}).

Theorems & Definitions (17)

• Theorem 1
• Definition 2
• Proposition 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Proposition 6
• proof