Majorization-type cluster robust bounds for block filters and eigensolvers

TL;DR

This paper develops majorization-based convergence bounds for block iterative methods used in eigenvalue problems and matrix filters, improving robustness and applicability to clustered eigenvalues and various algorithms.

Contribution

It introduces novel majorization-based bounds that enhance convergence analysis for block solvers and filters, accommodating eigenvalue clusters and broad algorithmic applications.

Findings

Bounds are robust with eigenvalue clusters.

Improves upon previous convergence bounds.

Applicable to multiple block iterative methods.

Abstract

Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and the closely related research on properties of matrix-based signal filters are challenging, and attract increasing attention due to their recent applications in spectral data clustering and graph-based signal processing. We combine majorization-based techniques pioneered for investigating the Rayleigh-Ritz method in [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521-1537] with tools of classical analysis of the block power method by Rutishauser [Numer. Math., 13 (1969), pp. 4-13] to derive convergence rate bounds of an abstract block iteration, wherein tuples of tangents of principal angles or relative errors of Ritz values are bounded using majorization in terms of arranged partial sums and tuples of convergence factors. Our novel bounds are robust in presence of clusters of eigenvalues, improve some previous results, and are applicable to most known block iterative solvers and matrix-based filters, e.g., to block power, Chebyshev, and Lanczos methods combined with shift-and-invert approaches and polynomial filtering.