On the Unreasonable Effectiveness of Single Vector Krylov Methods for Low-Rank Approximation

• Published in: arXiv.org 2023-11
• Main Authors: Meyer, Raphael A, Musco, Cameron, Musco, Christopher
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Eigenvalues; Mathematical analysis; Software; Subspace methods
• ISSN: 2331-8422

Krylov subspace methods are a ubiquitous tool for computing near-optimal 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 low-rank approximation. We address this gap between theory and practice by proving that small block Krylov methods essentially match all known low-rank approximation guarantees for large block methods. Via a black-box 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 matrix-vector products, single vector methods are essentially as effective as any choice of large block size. By combining our result with tail-bounds 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 low-rank approximation.