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

Technical Summary: A Replica Exchange MCMC Method for Disconnected Implicit Manifolds via Tubular Relaxation

1. Problem Statement

Standard Markov Chain Monte Carlo (MCMC) methods, specifically Constrained Hamiltonian Monte Carlo (CHMC), are designed to sample from probability measures supported on lower-dimensional manifolds defined by nonlinear constraints (e.g., fixed bond lengths in molecular dynamics or non-identifiable parameter sets in Bayesian inverse problems).

However, these methods face a fundamental limitation: topological disconnectedness. Most existing CHMC algorithms rely on local integrators (like RATTLE) that evolve dynamics within the local tangent space of the manifold. If the nonlinear constraints induce a manifold with multiple disconnected components, a local sampler becomes "trapped" within a single component, failing to explore the full support of the target distribution. This leads to biased statistical inference and a failure to capture the global geometry of the problem.

2. Methodology

The authors propose a novel Replica Exchange MCMC framework that bridges disconnected components by coupling a constrained chain with a relaxed auxiliary chain. The methodology is built on three pillars:

• Tubular Relaxation: Instead of enforcing a "hard" constraint ($\xi(q) = 0$), the authors introduce a "soft" relaxation. They define a family of auxiliary distributions $\pi_\epsilon$ in a tubular neighborhood of the manifold. This distribution penalizes deviations from the manifold using a quadratic term $\frac{1}{\epsilon}\|\xi(x)\|^2$. As $\epsilon \to 0$, this relaxed measure weakly converges to the true target measure $\mu$ supported on the manifold. Because the relaxed chain exists in the ambient Euclidean space, it can traverse the "gaps" between disconnected manifold components.
• Geometrically Informed Replica Exchange: To move information between the constrained chain (on the manifold $M$) and the relaxed chain (in the ambient space $\mathbb{R}^n$), the authors design a deterministic exchange mapping $S$. This mapping uses the normal bundle decomposition to swap the manifold component of the cold chain with the projected point of the hot chain, while preserving the normal displacement.
• Jacobian Correction: Because the exchange map $S$ is not measure-preserving, the authors derive a specific Metropolis-Hastings acceptance rule that accounts for the Jacobian determinant of the exchange map. To maintain computational efficiency, they propose a Gram determinant approximation, proving that the resulting bias in the total variation distance is of order $\mathcal{O}(\sqrt{\epsilon})$.

3. Key Contributions

• Theoretical Foundation: The paper provides rigorous proofs for the detailed balance, irreducibility, ergodicity, and convergence of the proposed sampler.
• Tubular Neighborhood Analysis: It establishes a formal mathematical link between the ambient space and the manifold using the properties of the normal bundle and the "reach" of the manifold.
• Approximation Error Bound: A significant contribution is the derivation of the error bound for the Gram approximation, showing that the method remains asymptotically exact as the relaxation parameter $\epsilon$ vanishes.
• Algorithmic Innovation: The introduction of a "reversible involution check" ensures that the numerical implementation of the exchange remains valid even when the projection onto the manifold is non-trivial.

4. Results and Validation

The effectiveness of the method was demonstrated through three diverse case studies:

• Synthetic Benchmark: In a 2D space with four disconnected ellipses, the proposed algorithm successfully transitioned between all components, whereas standard CHMC remained trapped in the component of its initialization. The authors also validated that the Gram approximation error scales as $\mathcal{O}(\sqrt{\epsilon})$.
• Infectious Disease Modeling: Applied to an SIR model with structural non-identifiability, the method successfully explored the disconnected parameter manifolds created by algebraic symmetries, correctly capturing the marginal distributions of non-identifiable parameters.
• Molecular Dynamics: Using a tetrahedral molecule model, the algorithm successfully sampled between two enantiomeric (mirror-image) configurations. It accurately recovered the theoretical probability ratio between the two disconnected components, a task impossible for standard constrained samplers.

5. Significance

This work provides a robust solution to a long-standing problem in geometric MCMC: global exploration of disconnected manifolds. By leveraging the geometry of tubular neighborhoods, the authors bridge the gap between constrained dynamics (which are efficient but local) and unconstrained dynamics (which are global but violate constraints). This has profound implications for:

• Computational Chemistry/Biology: Accurate sampling of molecular conformations and stereoisomers.
• Bayesian Statistics: Reliable uncertainty quantification in complex, non-identifiable inverse problems.
• Mathematical Physics: Sampling from configuration spaces with topological barriers.