Geometric Ergodicity for Hamiltonian Monte Carlo on Compact Manifolds

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.