Davis-Kahan Theorem under a moderate gap condition

TL;DR

This paper extends the Davis-Kahan theorem to cases with moderate eigenvalue gaps, providing a new perturbation bound that is effective when the perturbation is uncorrelated, potentially sharp up to a logarithmic factor.

Contribution

It introduces a novel bound for eigenspace perturbation under moderate gaps using a bootstrapping approach, applicable when perturbations are uncorrelated.

Findings

New bound for eigenspace perturbation with moderate gaps

Bound is efficient for uncorrelated perturbations

Bound is likely sharp up to a logarithmic term

Abstract

The classical Davis-Kahan theorem provides an efficient bound on the perturbation of eigenspaces of a matrix under a large (eigenvalue) gap condition. In this paper, we consider the case when the gap is moderate. Using a bootstrapping argument, we obtain a new bound which is efficient when the perturbation matrix is uncorrelated to the ground matrix. We believe that this bound is sharp up to a logarithmic term.