A polar-factor retraction on the symplectic Stiefel manifold with closed-form inverse

TL;DR

This paper introduces a new retraction on the symplectic Stiefel manifold with a closed-form inverse, improving computational efficiency and enabling better manifold-to-Euclidean mappings.

Contribution

It presents a novel retraction with a closed-form inverse on the symplectic Stiefel manifold, expanding the tools available for Riemannian computations.

Findings

The new retraction has comparable or better numerical performance than existing methods.

It is the first retraction with a closed-form inverse on this manifold besides the Cayley retraction.

Numerical experiments demonstrate efficiency gains in Riemannian applications.

Abstract

In Riemannian computing applications, it is crucial to map manifold data to a Euclidean domain, where vector space arithmetic is available, and back. Classical manifold theory guarantees the existence of such mappings, called charts and parameterizations, or, collectively, local coordinates. When computational efficiency is of the essence, practitioners usually resort to retraction maps to define local coordinates. Retractions yield first-order approximations of the Riemannian normal coordinates. This work introduces a new retraction on the symplectic Stiefel manifold that features a closed-form inverse. We expose essential features and compare the numerical performance to a selection of existing retractions. To the best of our knowledge, prior to this work, the so-called Cayley retraction was the only retraction on the symplectic Stiefel manifold with known closed-form inverse.