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- W4372272876 abstract "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." @default.
- W4372272876 created "2023-05-07" @default.
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- W4372272876 date "2023-05-04" @default.
- W4372272876 modified "2023-09-27" @default.
- W4372272876 title "On the Unreasonable Effectiveness of Single Vector Krylov Methods for Low-Rank Approximation" @default.
- W4372272876 doi "https://doi.org/10.48550/arxiv.2305.02535" @default.
- W4372272876 hasPublicationYear "2023" @default.
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