2022/01/12 by M. Zhou, Ming Zhou, Merico E. Argentati +7
Computer Science · Engineering · Mathematics · #65F15 #65N12 #65N25 #Algorithm #Block (permutation group theory) #Block matrix #Chebyshev filter #Cluster analysis #Combinatorics #Convergence (economics) #Eigenvalues and eigenvectors #FOS: Mathematics #Hermitian matrix #Iterative method #Lanczos resampling #Majorization #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Power iteration #Pure mathematics #Sparse and Compressive Sensing Techniques #Spectral clustering #cs.NA #math.NA #msc:65F15 #msc:65N12 #msc:65N25
paper · pdf · doi:10.48550/arxiv.2201.04517
24 pages, 2 figures
arxiv created 2022/01/12 · openalex publication_date 2022/01/12 · arxiv updated 2022/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.