# Riemannian optimization with a preconditioning scheme on the generalized Stiefel manifold

## TL;DR

This paper develops Riemannian optimization techniques with a preconditioning scheme on the generalized Stiefel manifold, improving computational efficiency and convergence for problems in science, engineering, and machine learning.

## Contribution

It introduces geometric components for Riemannian optimization with a non-standard metric on the generalized Stiefel manifold, enabling effective preconditioning.

## Key findings

- Preconditioning improves convergence speed.
- Geometric components are efficiently derived for the non-standard metric.
- Numerical experiments demonstrate the effectiveness of the approach.

## Abstract

Optimization problems on the generalized Stiefel manifold (and products of it) are prevalent across science and engineering. For example, in computational science they arise in symmetric (generalized) eigenvalue problems, in nonlinear eigenvalue problems, and in electronic structures computations, to name a few problems. In statistics and machine learning, they arise, for example, in various dimensionality reduction techniques such as canonical correlation analysis. In deep learning, regularization and improved stability can be obtained by constraining some layers to have parameter matrices that belong to the Stiefel manifold. Solving problems on the generalized Stiefel manifold can be approached via the tools of Riemannian optimization. However, using the standard geometric components for the generalized Stiefel manifold has two possible shortcomings: computing some of the geometric components can be too expensive and convergence can be rather slow in certain cases. Both shortcomings can be addressed using a technique called Riemannian preconditioning, which amounts to using geometric components derived by a precoditioner that defines a Riemannian metric on the constraint manifold. In this paper we develop the geometric components required to perform Riemannian optimization on the generalized Stiefel manifold equipped with a non-standard metric, and illustrate theoretically and numerically the use of those components and the effect of Riemannian preconditioning for solving optimization problems on the generalized Stiefel manifold.