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

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 see that for the fixed-rank problem, the bound for the power iteration is favourable in terms of simplicity and rounding error contribution. 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