Sharp Majorization-Type Cluster Robust Bounds for Block Filters and Eigensolvers

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1852-1878, December 2023.
Abstract. Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and closely related research on properties of matrix-based signal filters are challenging and are attracting increased 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 [A. V. Knyazev and M. E. Argentati, 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 sharp convergence rate bounds of abstract block iterations, 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 the presence of clusters of eigenvalues, improve 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 polynomial filtering. The sharpness of our bounds is fundamental, implying that the bounds cannot be improved without further assumptions.