Scaling of Radial Basis Functions

TL;DR

This paper investigates how the scale parameter affects the error and behavior of Radial Basis Function interpolation, revealing limitations and insights for selecting optimal scales in various function spaces.

Contribution

It provides a detailed analysis of the impact of scaling on RBF interpolation errors and bounds, including new insights into Sobolev spaces and the role of flat limits.

Findings

Error varies with kernel scale and space properties.

Flat limits and polynomial cases influence scale selection.

Standard error bounds can qualitatively track actual errors.

Abstract

This paper studies the influence of scaling on the behavior of Radial Basis Function interpolation. It focuses on certain central aspects, but does not try to be exhaustive. The most important questions are: How does the error of a kernel-based interpolant vary with the scale of the kernel chosen? How does the standard error bound vary? And since fixed functions may be in spaces that allow scalings, like global Sobolev spaces, is there a scale of the space that matches the function best? The last question is answered in the affirmative for Sobolev spaces, but the required scale may be hard to estimate. Scalability of functions turns out to be restricted for spaces generated by analytic kernels, unless the functions are band-limited. In contrast to other papers, polynomials and polyharmonics are included as flat limits when checking scales experimentally, with an independent computation. The numerical results show that the hunt for near-flat scales is questionable, if users include the flat limit cases right from the start. When there are not enough data to evaluate errors directly, the scale of the standard error bound can be varied, up to replacing the norm of the unknown function by the norm of the interpolant. This follows the behavior of the actual error qualitatively well, but is only of limited value for estimating error-optimal scales. For kernels and functions with unlimited smoothness, the given interpolation data are proven to be insufficient for determining useful scales.