A novel class of stabilized greedy kernel approximation algorithms: Convergence, stability & uniform point distribution

TL;DR

This paper introduces a new class of stabilized greedy kernel algorithms that improve the stability and convergence of meshfree function approximation, with proven optimal rates and better performance than existing methods.

Contribution

The paper develops and analyzes stabilized greedy kernel algorithms, providing convergence proofs and stability results, and demonstrates their superiority over traditional methods.

Findings

Algorithms achieve optimal stability and convergence rates.

Theoretical results extend to functions outside the native space.

Experimental results support the theoretical improvements.

Abstract

Kernel based methods provide a way to reconstruct potentially high-dimensional functions from meshfree samples, i.e., sampling points and corresponding target values. A crucial ingredient for this to be successful is the distribution of the sampling points. Since the computation of an optimal selection of sampling points may be an infeasible task, one promising option is to use greedy methods. Although these methods may be very effective, depending on the specific greedy criterion the chosen points might quickly lead to instabilities in the computation. To circumvent this problem, we introduce and investigate a new class of \textit{stabilized} greedy kernel algorithms, which can be used to create a scale of new selection strategies. We analyze these algorithms, and in particular we prove convergence results and quantify in a precise way the distribution of the selected points. These results allow to prove, in the case of certain Sobolev kernels, that the algorithms have optimal stability and optimal convergence rates, including for functions outside the native space of the kernel. The results also apply to the case of the usual $P$-greedy algorithm, significantly improving state-of-the-art results available in the literature. Illustrative experiments are presented that support the theoretical findings.