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.

翻译：近年来，文献中提倡使用随机化方法来加速数据科学与计算科学中出现的各类矩阵问题的求解。一种流行的随机化利用策略是将其作为降低问题规模的手段。然而，基于该策略的方法对于某些应用而言精度不足。随机化预处理是另一种利用随机化的方法，它能提供更高的精度。使用随机化预处理的主要挑战在于需要依赖底层迭代方法，因此迄今为止随机化预处理几乎只被应用于求解回归问题和线性系统。本文展示了如何将随机化预处理的适用范围扩展到数据科学中另一类重要问题：具有（广义）正交约束的优化问题。我们基于黎曼优化与黎曼预处理的框架，在计算主导典型相关以及Fisher线性判别分析问题上演示了我们的方法。针对这两个问题，我们评估了预处理对计算成本和渐近收敛性的影响，并通过实验验证了本方法的实用性。