Spectral alignment of kernel matrices and applications

TL;DR

This paper refines a multivariate Ingham-type theorem to derive new stability estimates for kernel matrices, especially for finitely smooth kernels like Matérn and Wendland, with implications for their eigenvectors.

Contribution

It introduces refined stability bounds for kernel matrices using an improved Ingham-type theorem, extending results to finitely smooth kernels and relating Rayleigh quotients across different smoothness levels.

Findings

Refined stability estimates for kernel matrices.

Relations between Rayleigh quotients for kernels of different smoothness.

Comments on eigenvector properties of kernel matrices.

Abstract

Kernel matrices are a key quantity in kernel-based approximation, and important properties such as stability and algorithmic convergence can be analyzed with their help. In this work we refine a multivariate Ingham-type theorem, which is then leveraged to obtain novel and refined stability estimates on kernel matrices. For this, we focus on the case of finitely smooth kernels, such as the family of Mat\'ern or Wendland kernels, while noting that the results also extend to norm-equivalent kernels. In particular we obtain results that relate the Rayleigh quotients of kernel matrices for kernels of different smoothness to each other. Finally we comment on conclusions for the eigenvectors of these kernel matrices.