Randomized Low Rank Matrix Approximation: Rounding Error Analysis and a Mixed Precision Algorithm

Connolly, Michael P. and Higham, Nicholas J. and Pranesh, Srikara (2022) Randomized Low Rank Matrix Approximation: Rounding Error Analysis and a Mixed Precision Algorithm. [MIMS Preprint]

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The available error bounds for randomized algorithms for computing a low rank approximation to a matrix assume exact arithmetic. Rounding errors potentially dominate the approximation error, though, especially when the algorithms are run in low precision arithmetic. We give a rounding error analysis of the method that computes a randomized rangefinder and then computes an approximate singular value decomposition approximation. Our analysis covers the basic method and the power iteration for the fixed-rank problem, as well as the power iteration for the fixed-precision problem. We give both worst-case and probabilistic rounding error bounds as functions of the problem dimensions and the rank. The worst-case bounds are pessimistic, but the probabilistic bounds are reasonably tight and still reliably bound the error in practice. We also propose a mixed precision version of the algorithm that offers potential speedups by gradually decreasing the precision during the execution of the algorithm.

Item Type: MIMS Preprint
Uncontrolled Keywords: randomized algorithms, low rank matrix approximation, singular value decomposition, rounding error analysis, probabilistic rounding error analysis, mixed precision algorithm
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 19 Jul 2022 11:23
Last Modified: 19 Jul 2022 11:23
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2863

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