The Geometric Foundations of Hamiltonian Monte Carlo

TL;DR

This paper develops a rigorous theoretical foundation for Hamiltonian Monte Carlo by constructing measures on smooth manifolds, which clarifies its efficiency and guides future improvements.

Contribution

It provides the first formal geometric framework for Hamiltonian Monte Carlo, enabling principled analysis and development of the algorithm.

Findings

Measures on smooth manifolds underpin HMC's efficiency

The theory guides the design of more effective implementations

Potential for new generalizations of HMC

Abstract

Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algorithm has in many ways impeded both principled developments of the method and use of the algorithm in practice. In this paper we develop the formal foundations of the algorithm through the construction of measures on smooth manifolds, and demonstrate how the theory naturally identifies efficient implementations and motivates promising generalizations.