Geodesic Monte Carlo on Embedded Manifolds

TL;DR

This paper advances Markov chain Monte Carlo methods by developing geodesic flow-based proposals for distributions on manifolds, enabling efficient sampling from complex geometric probability spaces.

Contribution

It introduces novel proposal mechanisms based on geodesic flows for distributions defined on manifolds, extending the geometric MCMC framework.

Findings

Developed geodesic flow proposals for hypersphere and Stiefel manifolds

Enhanced sampling efficiency for directional and matrix-valued distributions

Provided illustrative examples demonstrating the methods' applicability

Abstract

Markov chain Monte Carlo methods explicitly defined on the manifold of probability distributions have recently been established. These methods are constructed from diffusions across the manifold and the solution of the equations describing geodesic flows in the Hamilton--Jacobi representation. This paper takes the differential geometric basis of Markov chain Monte Carlo further by considering methods to simulate from probability distributions that themselves are defined on a manifold, with common examples being classes of distributions describing directional statistics. Proposal mechanisms are developed based on the geodesic flows over the manifolds of support for the distributions and illustrative examples are provided for the hypersphere and Stiefel manifold of orthonormal matrices.