Randomized Time Riemannian Manifold Hamiltonian Monte Carlo

Abstract

In the last decade several sampling methods have been proposed which rely on piecewise deterministic Markov processes (PDMPs). PDMPs are based on following deterministic trajectories with stochastic events which correspond to jumps in the state space. We propose implementing constraints in this setting to exploit geometries of high-dimensional problems by introducing a PDMP version of Riemannian manifold Hamiltonian Monte Carlo, which we call randomized time Riemannian manifold Hamiltonian Monte Carlo. Efficient sampling on constrained spaces is also needed in many applications including protein conformation modelling, directional statistics and free energy computations. We will show how randomizing the duration parameter for Hamiltonian flow can improve the robustness of Riemannian manifold Hamiltonian Monte Carlo methods. We will then compare methods on some example distributions which arise in application and provide an application of sampling on manifolds in high-dimensional covariance estimation.

Mixing time guarantees for unadjusted Hamiltonian Monte Carlo

“…In this work, we do not consider the corresponding Metropolis-adjusted HMC algorithm for sampling from the mean-field probability measure µ * , because the asymptotic bias due to the particle approximation cannot be eliminated by Metropolis-adjustment. Since we avoid Metropolis-adjustment, time step or duration adaptivity can be easily included in (2.13) -as in [41,8,48,42,78]. For promising alternative strategies to the particle approximation considered here, see [29,31].…”