  Recent literature has advocated the use of randomized methods for
accelerating the solution of various matrix problems arising throughout data
science and computational science. One popular strategy for leveraging
randomization is to use it as a way to reduce problem size. However, methods
based on this strategy lack sufficient accuracy for some applications.
Randomized preconditioning is another approach for leveraging randomization,
which provides higher accuracy. The main challenge in using randomized
preconditioning is the need for an underlying iterative method, thus randomized
preconditioning so far have been applied almost exclusively to solving
regression problems and linear systems. In this article, we show how to expand
the application of randomized preconditioning to another important set of
problems prevalent across data science: optimization problems with
(generalized) orthogonality constraints. We demonstrate our approach, which is
based on the framework of Riemannian optimization and Riemannian
preconditioning, on the problem of computing the dominant canonical
correlations and on the Fisher linear discriminant analysis problem. For both
problems, we evaluate the effect of preconditioning on the computational costs
and asymptotic convergence, and demonstrate empirically the utility of our
approach.
