31 research outputs found
Curvature and Concentration of Hamiltonian Monte Carlo in High Dimensions
In this article, we analyze Hamiltonian Monte Carlo (HMC) by placing it in
the setting of Riemannian geometry using the Jacobi metric, so that each step
corresponds to a geodesic on a suitable Riemannian manifold. We then combine
the notion of curvature of a Markov chain due to Joulin and Ollivier with the
classical sectional curvature from Riemannian geometry to derive error bounds
for HMC in important cases, where we have positive curvature. These cases
include several classical distributions such as multivariate Gaussians, and
also distributions arising in the study of Bayesian image registration. The
theoretical development suggests the sectional curvature as a new diagnostic
tool for convergence for certain Markov chains.Comment: Comments welcom