SIAM Journal on Optimization, Volume 31, Issue 2, Page 1546-1575, January 2021.
The symplectic Stiefel manifold, denoted by ${Sp}(2p,2n)$, 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. Optimization problems on ${Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra, and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on ${Sp}(2p,2n)$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on ${Sp}(2p,2n)$ akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods.

中文翻译：

---

辛 Stiefel 流形的黎曼优化

SIAM 优化杂志，第 31 卷，第 2 期，第 1546-1575 页，2021 年 1 月。
辛 Stiefel 流形，记作 ${Sp}(2p,2n)$，是标准辛空间 $\mathbb{R}^{2p}$ 和 $\mathbb{R}^{ 之间的线性辛映射集2n}$。当 $p=n$ 时，它简化为已知的 $2n\times 2n$ 辛矩阵集。${Sp}(2p,2n)$ 上的优化问题在各个领域都有应用，例如光学、量子物理学、数值线性代数和动力系统的模型降阶。本文的目的是提出和分析 ${Sp}(2p,2n)$ 上的梯度下降方法，其中梯度的概念源于黎曼度量。我们考虑 ${Sp}(2p,2n)$ 上的一种新颖的黎曼度量，类似于（标准）Stiefel 流形的规范度量。为了沿着反梯度执行可行的步骤，我们开发了两种类型的搜索策略：一种基于准测地线曲线，另一种基于辛凯莱变换。最终的优化算法被证明可以全局收敛到目标函数的临界点。数值实验说明了所提出方法的效率。