Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric

TL;DR

This paper explores the Riemannian geometry of the symplectic Stiefel manifold under the Euclidean metric, deriving geometric tools and applying them to optimization problems with demonstrated numerical effectiveness.

Contribution

It provides a detailed geometric analysis of the symplectic Stiefel manifold and develops Euclidean-based optimization algorithms for relevant problems.

Findings

Derived the Riemannian gradient for optimization.

Numerical experiments show effectiveness of the algorithms.

Analyzed the normal space and projections on the manifold.

Abstract

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space $\mathbb{R}^{2n\times 2p}$. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expression of the Riemannian gradient with respect to the Euclidean metric, which then used in optimization algorithms. Numerical experiments on the nearest symplectic matrix problem and the symplectic eigenvalue problem illustrate the effectiveness of Euclidean-based algorithms.