SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2994-3013, December 2023.
Abstract. We consider a Markov chain Monte Carlo method, known as Hamiltonian Monte Carlo (HMC), on compact manifolds in Euclidean space. It utilizes Hamiltonian dynamics to generate samples approximating a target distribution in high dimensions efficiently. The efficiency of HMC is characterized by its convergence property, called geometric ergodicity. This property is important to generate low-correlated samples. It also plays a crucial role in establishing the error estimate for the quadrature of bounded functions by HMC sampling, referred to as the Hoeffding-type inequality. While the geometric ergodicity has been proved for HMC on Euclidean space, it has not been established on manifolds. In this paper, we prove the geometric ergodicity for HMC on compact manifolds. As an example to confirm the efficiency of the proposed HMC method, we consider a sampling problem associated with the [math]-vortex problem on the unit sphere, which is a statistical model of two-dimensional turbulence. We apply HMC to approximate the statistical quantities with respect to the invariant measure of the [math]-vortex problem, called the Gibbs measure. We observe the organization of large vortex structures as seen in two-dimensional turbulence.

中文翻译：

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紧致流形上哈密顿蒙特卡罗的几何遍历性

《SIAM 数值分析杂志》，第 61 卷，第 6 期，第 2994-3013 页，2023 年 12 月。
摘要。我们考虑欧几里德空间中的紧流形上的马尔可夫链蒙特卡罗方法，称为哈密顿蒙特卡罗（HMC）。它利用哈密顿动力学有效地生成近似高维目标分布的样本。HMC 的效率由其收敛性（称为几何遍历性）来表征。此属性对于生成低相关样本非常重要。它还在通过 HMC 采样建立有界函数求积的误差估计（称为 Hoeffding 型不等式）方面发挥着至关重要的作用。虽然 HMC 的几何遍历性已在欧几里得空间上得到证明，但尚未在流形上得到证明。在本文中，我们证明了紧流形上 HMC 的几何遍历性。作为确认所提出的 HMC 方法效率的示例，我们考虑与单位球体上的[数学]涡问题相关的采样问题，这是二维湍流的统计模型。我们应用 HMC 来近似关于[数学]涡问题的不变测度（称为吉布斯测度）的统计量。我们观察二维湍流中大型涡旋结构的组织。