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Randomized block-Krylov subspace methods for low-rank approximation of matrix functions.

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Title: Randomized block-Krylov subspace methods for low-rank approximation of matrix functions.
Authors: Persson, David1 (AUTHOR) dpersson@flatironinstitute.org, Chen, Tyler2 (AUTHOR) tyler.chen@nyu.edu, Musco, Christopher2 (AUTHOR) cmusco@nyu.edu
Source: Linear Algebra & its Applications. Jul2026, Vol. 741, p32-65. 34p.
Subjects: Matrix functions, Krylov subspace, Singular value decomposition, Approximation theory, Matrix multiplications, Low-rank matrices, Numerical solutions for linear algebra
Abstract: The randomized SVD is a method to compute an inexpensive, yet accurate, low-rank approximation of a matrix. The algorithm assumes access to the matrix through matrix-vector products (matvecs). Therefore, when we would like to apply the randomized SVD to a matrix function, f (A) , one needs to approximate matvecs with f (A) using some other algorithm, which is typically treated as a black-box. Chen and Hallman (SIMAX 2023) argued that, in the common setting where matvecs with f (A) are approximated using Krylov subspace methods (KSMs), a more efficient low-rank approximation is possible if we open this black-box. They present an alternative approach that significantly outperforms the naive combination of KSMs with the randomized SVD, although the method lacked theoretical justification. In this work, we take a closer look at the method, and provide strong and intuitive error bounds that justify its excellent performance for low-rank approximation of matrix functions. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:The randomized SVD is a method to compute an inexpensive, yet accurate, low-rank approximation of a matrix. The algorithm assumes access to the matrix through matrix-vector products (matvecs). Therefore, when we would like to apply the randomized SVD to a matrix function, f (A) , one needs to approximate matvecs with f (A) using some other algorithm, which is typically treated as a black-box. Chen and Hallman (SIMAX 2023) argued that, in the common setting where matvecs with f (A) are approximated using Krylov subspace methods (KSMs), a more efficient low-rank approximation is possible if we open this black-box. They present an alternative approach that significantly outperforms the naive combination of KSMs with the randomized SVD, although the method lacked theoretical justification. In this work, we take a closer look at the method, and provide strong and intuitive error bounds that justify its excellent performance for low-rank approximation of matrix functions. [ABSTRACT FROM AUTHOR]
ISSN:00243795
DOI:10.1016/j.laa.2026.03.029