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Asymptotic expansions for spectral convergence of compact self-adjoint operators on general spectral subsets, with application to kernel Gram matrices

Eunseong Bae, Wolfgang Polonik

TL;DR

This work develops a general perturbation framework for compact self-adjoint operators to obtain asymptotic expansions of eigenvalues and eigenprojections indexed by a general subset $\mathcal{J}$ of the spectrum, under operator-norm perturbations. The framework distinguishes well-separated and clustered eigenvalues and provides explicit finite-sample expansions and remainder bounds via off-diagonal perturbations and cluster-specific matrices. As an application, the authors translate these results to kernel Gram matrices by embedding the problem in the RKHS associated with a Mercer kernel, establishing finite-sample concentration inequalities and weak convergence results for eigenvalues and eigenprojections based on kernel assumptions that are easy to verify. The approach yields contracts and limit results that extend classical perturbation theory to statistically perturbed, data-driven operators, with practical implications for kernel-based methods in statistics and machine learning, including spectral analysis, PCA, and kernel learning. Overall, the paper offers a cohesive theory linking operator perturbation, spectral convergence, and kernel methods under accessible kernel conditions, enabling precise asymptotics and probabilistic guarantees.

Abstract

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space under operator norm perturbations, and derive asymptotic expansions for their eigenvalues and eigenprojections. Our analysis focuses on eigenvalues indexed by a general subset, with minimal restrictions on their selection. The usefulness of the provided expansions is illustrated by an application to kernel Gram matrices, deriving concentration inequalities as well as weak convergence results, which, in contrast to existing literature, are primarily relying on assumptions on the kernel that are easy to check.

Asymptotic expansions for spectral convergence of compact self-adjoint operators on general spectral subsets, with application to kernel Gram matrices

TL;DR

This work develops a general perturbation framework for compact self-adjoint operators to obtain asymptotic expansions of eigenvalues and eigenprojections indexed by a general subset of the spectrum, under operator-norm perturbations. The framework distinguishes well-separated and clustered eigenvalues and provides explicit finite-sample expansions and remainder bounds via off-diagonal perturbations and cluster-specific matrices. As an application, the authors translate these results to kernel Gram matrices by embedding the problem in the RKHS associated with a Mercer kernel, establishing finite-sample concentration inequalities and weak convergence results for eigenvalues and eigenprojections based on kernel assumptions that are easy to verify. The approach yields contracts and limit results that extend classical perturbation theory to statistically perturbed, data-driven operators, with practical implications for kernel-based methods in statistics and machine learning, including spectral analysis, PCA, and kernel learning. Overall, the paper offers a cohesive theory linking operator perturbation, spectral convergence, and kernel methods under accessible kernel conditions, enabling precise asymptotics and probabilistic guarantees.

Abstract

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space under operator norm perturbations, and derive asymptotic expansions for their eigenvalues and eigenprojections. Our analysis focuses on eigenvalues indexed by a general subset, with minimal restrictions on their selection. The usefulness of the provided expansions is illustrated by an application to kernel Gram matrices, deriving concentration inequalities as well as weak convergence results, which, in contrast to existing literature, are primarily relying on assumptions on the kernel that are easy to check.
Paper Structure (30 sections, 25 theorems, 181 equations)

This paper contains 30 sections, 25 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic expansions for spectral convergence of compact self-adjoint operators
  4. Setting
  5. Asymptotic expansions for eigenprojections of compact self-adjoint operators
  6. Asymptotic expansions for eigenvalues of compact self-adjoint operators
  7. Case 1: Well-separated eigenvalues
  8. Case 2: Clustered eigenvalues
  9. Asymptotic expansions for spectral convergence of kernel Gram matrices
  10. Setting
  11. Covariance operators in Reproducing kernel Hilbert spaces
  12. Asymptotic expansions for eigenprojections of kernel Gram matrices
  13. Asymptotic expansions for eigenvalues of kernel Gram matrices
  14. Possible extension to random index sets
  15. Proofs of Section \ref{['spec']}
  16. ...and 15 more sections

Key Result

Theorem 3.3

Suppose Assumptions spec:assump1 and spec:assump2 hold, and $\frac{\|\hat{\mathcal{H}} - \mathcal{H}\|_{{\operatorname{op}},\mathbb{H}}}{\gamma_{\raisebox{-2pt}{$\space \mathcal{J}$}}} < \frac{1}{4}.$ Then, we have where

Theorems & Definitions (55)

  • Theorem 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Lemma 4.2
  • Proposition 4.3
  • Remark 4.5
  • Lemma 4.6
  • Theorem 4.7
  • ...and 45 more