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2014 NeurIPS NeurIPS 2014

Positive Curvature and Hamiltonian Monte Carlo

Abstract

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.

🌉 Interdisciplinary Bridge - Artificial Intelligence and Mathematics & Optimization

🐣 Hot Topic Early Bird - markov chain

Authors

Christof Seiler, Simon Rubinstein-Salzedo, Susan Holmes

Topics

Artificial Intelligence > Bayesian & Probabilistic > Bayesian Learning Machine Learning > Optimization & Theory > Bayesian Inference Mathematics & Optimization > Mathematics > Geometry Mathematics & Optimization > Optimization > Stochastic Methods Artificial Intelligence > Bayesian & Probabilistic > Bayesian Inference Machine Learning > Bayesian & Probabilistic > Markov Chain Monte Carlo

Keywords

markov chain monte carlo markov chain riemannian geometry hamiltonian monte carlo riemannian manifold sectional curvature jacobi metric

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