Hamiltonian Monte Carlo on Symmetric and Homogeneous Spaces via Symplectic Reduction

TL;DR

This paper develops a framework for Hamiltonian Monte Carlo on manifolds by leveraging symplectic reduction, enabling efficient sampling on complex spaces like spheres, hyperbolic spaces, and matrix manifolds.

Contribution

It introduces a method to construct Hamiltonian systems on homogeneous spaces via symplectic reduction, facilitating HMC on manifolds with global coordinates.

Findings

Enables HMC on a wide class of manifolds.

Provides a unified geometric framework.

Simplifies computations on complex manifolds.

Abstract

The Hamiltonian Monte Carlo method generates samples by introducing a mechanical system that explores the target density. For distributions on manifolds it is not always simple to perform the mechanics as a result of the lack of global coordinates, the constraints of the manifold, and the requirement to compute the geodesic flow. In this paper we explain how to construct the Hamiltonian system on naturally reductive homogeneous spaces using symplectic reduction, which lifts the HMC scheme to a matrix Lie group with global coordinates and constant metric. This provides a general framework that is applicable to many manifolds that arise in applications, such as hyperspheres, hyperbolic spaces, symmetric positive-definite matrices, Grassmannian, and Stiefel manifolds.