Unperturbed: spectral analysis beyond Davis-Kahan

TL;DR

This paper develops refined spectral perturbation bounds that account for the nature of perturbations, leading to tighter results especially for random perturbations, and demonstrates their effectiveness in community detection within stochastic blockmodels.

Contribution

It introduces new spectral perturbation bounds that improve classical results by considering perturbation structure, enabling exact community recovery in sparse stochastic blockmodels.

Findings

Tighter bounds for spectral perturbations in random settings.

Successful exact community detection in sparse stochastic blockmodels.

Enhanced analysis of clustering algorithms using new perturbation theory.

Abstract

Classical matrix perturbation results, such as Weyl's theorem for eigenvalues and the Davis-Kahan theorem for eigenvectors, are general purpose. These classical bounds are tight in the worst case, but in many settings sub-optimal in the typical case. In this paper, we present perturbation bounds which consider the nature of the perturbation and its interaction with the unperturbed structure in order to obtain significant improvements over the classical theory in many scenarios, such as when the perturbation is random. We demonstrate the utility of these new results by analyzing perturbations in the stochastic blockmodel where we derive much tighter bounds than provided by the classical theory. We use our new perturbation theory to show that a very simple and natural clustering algorithm -- whose analysis was difficult using the classical tools -- nevertheless recovers the communities of the blockmodel exactly even in very sparse graphs.