Perturbation bounds for eigenspaces under a relative gap condition

TL;DR

This paper develops new bounds for how small perturbations affect eigenspaces of self-adjoint operators, especially under structured random perturbations and a relative gap condition, using a novel contraction approach.

Contribution

It introduces a new contraction-based method for spectral perturbation bounds under a relative gap condition, improving understanding of eigenspace stability for structured random perturbations.

Findings

Sharp bounds for empirical covariance operator eigenspaces

Effective bounds under a relative gap condition

Contrasts with previous spectral perturbation methods

Abstract

A basic problem in operator theory is to estimate how a small perturbation effects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, taylored for structured random perturbations. As a main example, we consider the empirical covariance operator, and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches.