A Replica Exchange Markov Chain Monte Carlo Method for Disconnected Implicit Manifolds via Tubular Relaxation
For practitioners using constrained MCMC, this method addresses the limitation of existing algorithms that assume connected manifolds, enabling sampling from a broader class of implicit manifolds.
The paper proposes a replica exchange MCMC method that couples a constrained chain on an implicit manifold with a relaxed auxiliary chain in a tubular neighborhood, enabling sampling from disconnected manifolds. The method is proven to satisfy detailed balance, irreducibility, ergodicity, and convergence, and is demonstrated on molecular and biological systems.
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.