Irreversible Langevin MCMC on Lie Groups

Alexis Arnaudon, Alessandro Barp, So Takao

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 $$\mathcal G= SO(3)$$ .