Spectral convergence of empirical integral operators with discontinuous kernels

TL;DR

This paper investigates how empirical integral operators with possibly discontinuous kernels converge spectrally to their continuous versions as sample size increases, providing explicit convergence rates.

Contribution

It extends spectral convergence analysis to kernels that are not necessarily positive or continuous, with explicit rates of convergence.

Findings

Empirical operators converge to continuous operators as sample size grows.

Convergence rates are explicitly derived for kernels with relaxed assumptions.

Spectral properties are preserved under these relaxed conditions.

Abstract

We study the spectral behavior as the sample size $n \to +\infty$ of integral operators defined by convolution of a non-negative symmetric kernel k with respect to empirical measures $\mu_n = \frac{1}{n} \sum_{i=1}^n \delta_{X_i}$, where $\{X_i\}_{i=1}^n$ are independent uniform samples from a compact probability metric space $(\mathcal{X},d,\mu)$. Relaxing the usual positivity and continuity assumptions on k, we prove the convergence of these empirical operators to their continuous counterparts, and provide explicit convergence rates.