Spectral properties of kernel matrices in the flat limit

TL;DR

This paper investigates the spectral properties of kernel matrices in the flat limit, deriving asymptotic expressions for eigenvalues and eigenvectors, and linking them to orthogonal polynomials, for both smooth and finitely smooth kernels.

Contribution

It provides new asymptotic formulas for kernel matrix eigenvalues and eigenvectors in the flat limit, connecting spectral properties to orthogonal polynomials.

Findings

Asymptotic expressions for determinants of kernel matrices

Eigenvalue asymptotics in the flat limit

Eigenvector expressions linked to orthogonal polynomials

Abstract

Kernel matrices are of central importance to many applied fields. In this manuscript, we focus on spectral properties of kernel matrices in the so-called ``flat limit'', which occurs when points are close together relative to the scale of the kernel. We establish asymptotic expressions for the determinants of the kernel matrices, which we then leverage to obtain asymptotic expressions for the main terms of the eigenvalues. Analyticity of the eigenprojectors yields expressions for limiting eigenvectors, which are strongly tied to discrete orthogonal polynomials. Both smooth and finitely smooth kernels are covered, with stronger results available in the finite smoothness case.