Hybrid Monte Carlo methods for sampling probability measures on submanifolds

TL;DR

This paper develops and analyzes generalized Hybrid Monte Carlo algorithms for sampling probability measures on submanifolds, emphasizing the importance of a reverse projection check to ensure unbiased sampling and reversibility.

Contribution

It extends GHMC methods with a reverse projection check for submanifold sampling, providing a rigorous analysis and demonstrating its effectiveness through numerical experiments.

Findings

Reverse projection check enforces reversibility for large timesteps

The scheme reduces bias in invariant measure sampling

Numerical experiments confirm the theoretical advantages

Abstract

Probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. We show here how to generalize to GHMC a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for large timesteps and hence avoid biases in the invariant measure. We also provide a full mathematical analysis of such procedures, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples.