# Irreversible Langevin MCMC on Lie Groups

## TL;DR

This paper introduces an irreversible MCMC algorithm tailored for Lie groups, leveraging geometric structures to enhance convergence speed, and demonstrates its effectiveness with the example of SO(3).

## Contribution

It develops a novel irreversible Hamiltonian Monte Carlo-like method on Lie groups using geometric and dissipation structures, generalizing HMC.

## Key findings

- Algorithm converges faster than reversible methods.
- Recovers HMC as a special case with long OU process simulation.
- Numerical experiments on SO(3) validate the approach.

## Abstract

It is well-known that irreversible MCMC algorithms converge faster to their stationary distributions than reversible ones. Using the special geometric structure of Lie groups $\mathcal G$ and dissipation fields compatible with the symplectic structure, we construct an irreversible HMC-like MCMC algorithm on $\mathcal G$, where we first update the momentum by solving an OU process on the corresponding Lie algebra $\mathfrak g$, and then approximate the Hamiltonian system on $\mathcal G \times \mathfrak g$ with a reversible symplectic integrator followed by a Metropolis-Hastings correction step. In particular, when the OU process is simulated over sufficiently long times, we recover HMC as a special case. We illustrate this algorithm numerically using the example $\mathcal G = SO(3)$.