Robust Monte Carlo Sampling using Riemannian Nosé-Poincaré Hamiltonian Dynamics

We present a Monte Carlo sampler using a modified Nose-Poincare Hamiltonian along with Riemannian preconditioning. Hamiltonian Monte Carlo samplers allow better exploration of the state space as opposed to random walk-based methods, but, from a molecular dynamics perspective, may not necessarily provide samples from the canonical ensemble. Nose-Hoover samplers rectify that shortcoming, but the resultant dynamics are not Hamiltonian. Furthermore, usage of these algorithms on large real-life datasets necessitates the use of stochastic gradients, which acts as another potentially destabilizing source of noise. In this work, we propose dynamics based on a modified Nose-Poincare Hamiltonian augmented with Riemannian manifold corrections. The resultant symplectic sampling algorithm samples from the canonical ensemble while using structural cues from the Riemannian preconditioning matrices to efficiently traverse the parameter space. We also propose a stochastic variant using additional terms in the Hamiltonian to correct for the noise from the stochastic gradients. We show strong performance of our algorithms on synthetic datasets and high-dimensional Poisson factor analysis-based topic modeling scenarios.