EAGER: Numerical Accuracy of Randomized Algorithms for Matrix Multiplication and Least Squares

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The PI proposes to investigate the numerical accuracy and robustness of randomized algorithms for matrix multiplication and overdetermined least squares problems. Existing analyses of randomized algorithms are mostly concerned with asymptotic time and space complexity in exact arithmetic, and very little is known about their numerical behavior in floating point arithmetic. The PI proposes to develop a numerical perturbation and stability theory for randomized algorithms for matrix multiplication and least squares problems. This entails the invention of new approaches and concepts to capture the numerical behavior of randomized algorithms. It is not at all clear what ``numerical stability'' means in this context, let alone how it should be defined. How does one distinguish variability caused by randomization from variability caused by finite precision? Where should parameters like failure probability, choice of probabilities, and amount of sampling be accounted for? Proposed approaches for answering these questions will include matrix perturbation analysis, probability theory, and methods on matrix manifolds. Extensive numerical experiments will be performed to corroborate the analyses.The motivation for randomized algorithms is the need for streaming massive data sets that are too large for traditional deterministic algorithms. Randomized algorithms have been implemented successfully for applications such as pattern recognition, social network analysis, population genetics, circuit testing, and text classification. The proposed research will help to determine for which application domains a randomized algorithm is suitable, and it will also result in practical bounds and recommendations for parameter choices to achieve a user-specified accuracy. The proposed research is highly relevant because randomized algorithms will be indispensable for exascale computing, in applications like high energy physics and astronomy, where peta bytes of data are expected to stream in daily and tasks like rare event detection make it imperative to have a good understanding of numerical accuracy and robustness.