A Replica Exchange Markov Chain Monte Carlo Method for Disconnected Implicit Manifolds via Tubular Relaxation

TL;DR

This paper introduces a replica exchange MCMC method that efficiently samples from disconnected implicit manifolds by coupling a constrained chain with a relaxed auxiliary chain, overcoming limitations of existing methods.

Contribution

The authors develop a novel replica exchange framework that enables sampling from disconnected manifolds, extending constrained Hamiltonian Monte Carlo techniques to complex geometric structures.

Findings

The proposed algorithm satisfies detailed balance, irreducibility, and ergodicity.

It effectively samples from disconnected manifolds in molecular and biological systems.

The method broadens the applicability of constrained MCMC methods.

Abstract

Markov chain Monte Carlo (MCMC) methods provide powerful framework for sampling unknown probability measures across a wide range of scientific applications. In some settings, the target distribution is supported on a lower-dimensional submanifold of Euclidean space defined by nonlinear constraints, motivating the development of constrained Hamiltonian Monte Carlo (CHMC) methods. Most existing CHMC algorithms rely on the assumption that the implicit manifold is connected, allowing local constrained integrators such as RATTLE to explore the posterior ergodically. In practice, this assumption is occasionally violated due to complex geometric structures induced by nonlinear constraints of a model. We propose a replica exchange MCMC framework that couples a constrained chain evolving on the implicit manifold with a relaxed auxiliary chain defined in a tubular neighborhood of the constraint. The relaxed chain enables transitions between disconnected components. We show that the resulting algorithm enables sampling from a broader class of implicit manifolds, including those with disconnected components. We prove that the proposed sampler satisfies detailed balance, irreducibility, ergodicity, and convergence. We also demonstrate its effectiveness on examples from molecular and biological dynamical systems.