Sharp error bounds for Ritz vectors and approximate singular vectors

TL;DR

This paper presents improved, sharp error bounds for Ritz vectors and approximate singular vectors in symmetric eigenvalue problems, surpassing classical bounds and applicable to various related problems.

Contribution

It introduces novel bounds that significantly improve classical error estimates for Ritz vectors, invariant subspaces, and singular vectors, with broad applicability.

Findings

Bounds outperform Davis-Kahan $ ext{sin} heta$ theorem

Applicable to invariant subspaces and singular vectors

Effective even when classical bounds suggest no accuracy

Abstract

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan $\sin\theta$ theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the $\sin\theta$ theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space.