Kernel-based interpolation at approximate Fekete points

TL;DR

This paper introduces a method for constructing approximate Fekete points for kernel interpolation by maximizing a truncated kernel Gram matrix determinant, providing uniform error estimates and demonstrating super-exponential convergence for Gaussian kernels.

Contribution

It presents a novel approach to approximate Fekete point construction using determinant maximization and offers convergence analysis with numerical validation.

Findings

Approximate Fekete points are obtained via convex optimization for Gaussian kernels.

Uniform error estimates are established for kernel interpolants at these points.

Super-exponential convergence is demonstrated for Gaussian kernel interpolation.

Abstract

We construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion of the kernel. Uniform error estimates are proved for kernel interpolants at the resulting points. If the kernel is Gaussian we show that the approximate Fekete points in one dimension are the solution to a convex optimisation problem and that the interpolants converge with a super-exponential rate. Numerical examples are provided for the Gaussian kernel.