A useful variant of the Davis--Kahan theorem for statisticians

TL;DR

This paper introduces a more natural and convenient variant of the Davis--Kahan theorem for statisticians, relaxing eigenvalue separation conditions and extending to asymmetric and non-square matrices, improving bounds in statistical analyses.

Contribution

It presents a new variant of the Davis--Kahan theorem based solely on population eigenvalue separation, applicable to asymmetric and non-square matrices, enhancing its utility in statistical contexts.

Findings

The new variant depends only on population eigenvalue separation.

Extension to asymmetric and non-square matrices with singular vectors.

Improved bounds in certain statistical applications.

Abstract

The Davis--Kahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions. It relies on an eigenvalue separation condition between certain relevant population and sample eigenvalues. We present a variant of this result that depends only on a population eigenvalue separation condition, making it more natural and convenient for direct application in statistical contexts, and improving the bounds in some cases. We also provide an extension to situations where the matrices under study may be asymmetric or even non-square, and where interest is in the distance between subspaces spanned by corresponding singular vectors.