论文标题
对符号齿状歧管的利曼式优化
Riemannian Optimization on the Symplectic Stiefel Manifold
论文作者
论文摘要
Symplectic Stiefel歧管，由$ \ Mathrm {sp}（2p，2n）$表示，是标准符号空间之间的一组线性符号映射，$ \ mathbb {r}^{2p} {2p} $和$ \ Mathbb {r}^r}^^r}^^2n} $。当$ p = n $时，它将减少到众所周知的$ 2N \ times 2n $ symplectic矩阵。 $ \ Mathrm {sp}（2p，2n）上的优化问题在各个区域中找到应用程序，例如光学，量子物理，数值线性代数和模型订单降低动态系统。本文的目的是在$ \ mathrm {sp}（2p，2n）$上提出和分析梯度降低的方法，其中梯度的概念源自Riemannian指标。我们考虑了$ \ mathrm {sp}（2p，2n）$类似于（标准）stiefel歧管的规范指标的新颖的Riemannian指标。为了沿着抗移民执行可行的步骤，我们制定了两种类型的搜索策略：一个基于准晶格曲线，另一个基于符号cayley转换。事实证明，所得的优化算法在全球范围内收敛到目标函数的关键点。数值实验说明了提出的方法的效率。
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. Optimization problems on $\mathrm{Sp}(2p,2n)$ 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 $\mathrm{Sp}(2p,2n)$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on $\mathrm{Sp}(2p,2n)$ 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.
