Positive Curvature and Hamiltonian Monte Carlo

Part of Advances in Neural Information Processing Systems 27 (NIPS 2014)

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Authors

Christof Seiler, Simon Rubinstein-Salzedo, Susan Holmes

Abstract

<p>The Jacobi metric introduced in mathematical physics can be used to analyze Hamiltonian Monte Carlo (HMC). In a geometrical setting, each step of HMC corresponds to a geodesic on a Riemannian manifold with a Jacobi metric. Our calculation of the sectional curvature of this HMC manifold allows us to see that it is positive in cases such as sampling from a high dimensional multivariate Gaussian. We show that positive curvature can be used to prove theoretical concentration results for HMC Markov chains.</p>