A polar-factor retraction on the symplectic Stiefel manifold with closed-form inverse
For researchers in Riemannian optimization and manifold learning, this provides a new computational tool with a closed-form inverse, though it is an incremental addition to existing retractions.
The paper introduces a new retraction on the symplectic Stiefel manifold with a closed-form inverse, addressing a computational need in Riemannian computing. Numerical comparisons show it performs competitively with existing retractions, including the Cayley retraction.
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.