Convergence rate of the data-independent $P$-greedy algorithm in kernel-based approximation

TL;DR

This paper proves the near-optimal convergence rate of the data-independent P-greedy algorithm in kernel-based function approximation, demonstrating its effectiveness and asymptotic uniformity in Sobolev spaces.

Contribution

It provides the first rigorous proof of the convergence rate for the P-greedy algorithm, linking it to greedy algorithms in reduced basis methods and confirming asymptotic uniformity.

Findings

Convergence rate is near-optimal for Sobolev space kernels.

Selected points by the algorithm are asymptotically uniformly distributed.

The proof connects kernel approximation with reduced basis greedy algorithms.

Abstract

Kernel-based methods provide flexible and accurate algorithms for the reconstruction of functions from meshless samples. A major question in the use of such methods is the influence of the samples locations on the behavior of the approximation, and feasible optimal strategies are not known for general problems. Nevertheless, efficient and greedy point-selection strategies are known. This paper gives a proof of the convergence rate of the data-independent \textit{$P$-greedy} algorithm, based on the application of the convergence theory for greedy algorithms in reduced basis methods. The resulting rate of convergence is shown to be near-optimal in the case of kernels generating Sobolev spaces. As a consequence, this convergence rate proves that, for kernels of Sobolev spaces, the points selected by the algorithm are asymptotically uniformly distributed, as conjectured in the paper where the algorithm has been introduced.