Faster Randomized Methods for Orthogonality Constrained Problems

TL;DR

This paper introduces an extension of randomized preconditioning techniques to orthogonality constrained optimization problems, improving efficiency and accuracy in data science applications like canonical correlations and Fisher analysis.

Contribution

It expands randomized preconditioning to Riemannian optimization problems, providing a new approach for faster solutions with higher accuracy.

Findings

Preconditioning reduces computational costs.

Preconditioning improves convergence rates.

Empirical results confirm utility in data science tasks.

Abstract

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.