Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric

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.

References in corpus (6)

• On symplectic eigenvalues of positive definite matrices
• Optimal Control Theory for Continuous Variable Quantum Gates
• Riemannian Optimization on the Symplectic Stiefel Manifold
• Derivatives of symplectic eigenvalues and a Lidskii type theorem
• Critical Topology for Optimization on the Symplectic Group
• Optimization on the symplectic group

Cited by in corpus (3)

• Constraint optimization and $\mathcal{S}\mathcal{U}(N)$ quantum control landscapes
• A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold
• A generalized canonical metric for optimization on the indefinite Stiefel manifold