On the Unreasonable Effectiveness of Single Vector Krylov Methods for LowRank Approximation
Abstract
Krylov subspace methods are a ubiquitous tool for computing nearoptimal rank $k$ approximations of large matrices. While "large block" Krylov methods with block size at least $k$ give the best known theoretical guarantees, block size one (a single vector) or a small constant is often preferred in practice. Despite their popularity, we lack theoretical bounds on the performance of such "small block" Krylov methods for lowrank approximation. We address this gap between theory and practice by proving that small block Krylov methods essentially match all known lowrank approximation guarantees for large block methods. Via a blackbox reduction we show, for example, that the standard single vector Krylov method run for $t$ iterations obtains the same spectral norm and Frobenius norm error bounds as a Krylov method with block size $\ell \geq k$ run for $O(t/\ell)$ iterations, up to a logarithmic dependence on the smallest gap between sequential singular values. That is, for a given number of matrixvector products, single vector methods are essentially as effective as any choice of large block size. By combining our result with tailbounds on eigenvalue gaps in random matrices, we prove that the dependence on the smallest singular value gap can be eliminated if the input matrix is perturbed by a small random matrix. Further, we show that single vector methods match the more complex algorithm of [Bakshi et al. `22], which combines the results of multiple block sizes to achieve an improved algorithm for Schatten $p$norm lowrank approximation.
 Publication:

arXiv eprints
 Pub Date:
 May 2023
 DOI:
 10.48550/arXiv.2305.02535
 arXiv:
 arXiv:2305.02535
 Bibcode:
 2023arXiv230502535M
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Numerical Analysis;
 65F55 (Primary) 65F15 (Secondary);
 G.1.3;
 F.2.1
 EPrint:
 41 pages, 7 figures