Evaluating the Implicit Midpoint Integrator for Riemannian Manifold Hamiltonian Monte Carlo

Area of Science:

• Computational mathematics
• Statistical physics
• Numerical analysis

Background:

• Hamiltonian Monte Carlo (HMC) is a widely used Markov chain Monte Carlo (MCMC) method.
• The generalized leapfrog integrator is the standard for HMC on Riemannian manifolds.
• Alternative integrators may offer improved performance or properties.

Purpose of the Study:

• To evaluate the implicit midpoint integrator as an alternative to the generalized leapfrog integrator for HMC.
• To analyze the theoretical properties of the implicit midpoint integrator in this context.
• To empirically assess its performance regarding energy conservation, volume preservation, and reversibility.

Main Methods:

• Theoretical analysis of the implicit midpoint integrator's properties.
• Empirical comparison with the generalized leapfrog integrator using key performance metrics.
• Assessment of energy conservation, volume preservation, and reversibility.

Main Results:

• The implicit midpoint integrator exhibits better energy conservation compared to the leapfrog integrator.
• This improved energy conservation leads to higher acceptance rates in the Markov chain.
• The implicit midpoint integrator also demonstrates superior volume preservation and reversibility.

Conclusions:

• The implicit midpoint integrator is a viable and potentially more accurate alternative for HMC on Riemannian manifolds.
• Despite slower iteration times, its superior conservation properties can yield a more accurate sampling procedure.
• Further research into alternative integrators for HMC is warranted.