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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Abstract
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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