Hamiltonian Monte Carlo on Symmetric and Homogeneous Spaces via\n Symplectic Reduction

2019/03/06 by Alessandro Barp, Barp, Alessandro, A.D. Kennedy +3
Chemistry · Computer Science · Mathematics · #Advanced Algebra and Geometry #Computation (stat.CO) #FOS: Computer and information sciences #Geometry and complex manifolds #Markov Chains and Monte Carlo Methods #Molecular spectroscopy and chirality #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1903.02699

openalex publication_date 2019/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Hamiltonian Monte Carlo method generates samples by introducing a\nmechanical system that explores the target density. For distributions on\nmanifolds it is not always simple to perform the mechanics as a result of the\nlack of global coordinates, the constraints of the manifold, and the\nrequirement to compute the geodesic flow. In this paper we explain how to\nconstruct the Hamiltonian system on naturally reductive homogeneous spaces\nusing symplectic reduction, which lifts the HMC scheme to a matrix Lie group\nwith global coordinates and constant metric. This provides a general framework\nthat is applicable to many manifolds that arise in applications, such as\nhyperspheres, hyperbolic spaces, symmetric positive-definite matrices,\nGrassmannian, and Stiefel manifolds.\n

Citations

• Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
• A Conceptual Introduction to Hamiltonian Monte Carlo

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