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

Raphael Meyer, Cameron Musco, Christopher Musco

    Research output: Contribution to conferencePaperpeer-review


    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 ℓ ≥ k run for O(t/ℓ) 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.

    Original languageEnglish (US)
    Number of pages35
    StatePublished - 2024
    Event35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States
    Duration: Jan 7 2024Jan 10 2024


    Conference35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024
    Country/TerritoryUnited States

    ASJC Scopus subject areas

    • Software
    • General Mathematics


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