# The Geometric Foundations of Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/1410.5110 · HTML: https://mlsanity.com/paper/1410.5110/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2014.

## Built on by (most cited first, 19 of 19)

- [MCMC Variational Inference via Uncorrected Hamiltonian Annealing](https://mlsanity.com/paper/2107.04150/) (2021)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [A Survey of Monte Carlo Methods for Parameter Estimation](https://mlsanity.com/paper/2107.11820/) (2021)
- [Energy based diffusion generator for efficient sampling of Boltzmann distributions](https://mlsanity.com/paper/2401.02080/) (2024)
- [Second Order Ensemble Langevin Method for Sampling and Inverse Problems](https://mlsanity.com/paper/2208.04506/) (2022)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers](https://mlsanity.com/paper/2307.03460/) (2023)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)
- [Variational Representations of Annealing Paths: Bregman Information under Monotonic Embedding](https://mlsanity.com/paper/2209.07481/) (2022)
- [History-Based, Bayesian, Closure for Stochastic Parameterization: Application to Lorenz '96](https://mlsanity.com/paper/2210.14488/) (2022)
- [Uncertainty quantification of neural network models of evolving processes via Langevin sampling](https://mlsanity.com/paper/2504.14854/) (2025)
- [Hamiltonian Monte Carlo with Asymmetrical Momentum Distributions](https://mlsanity.com/paper/2110.12907/) (2021)
- [Staying the course: Locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds](https://mlsanity.com/paper/2204.10413/) (2022)
- [Hamiltonian Monte Carlo on ReLU Neural Networks is Inefficient](https://mlsanity.com/paper/2410.22065/) (2024)
- [Counterdiabatic Hamiltonian Monte Carlo](https://mlsanity.com/paper/2602.21272/) (2026)
- [The Application of Zig-Zag Sampler in Sequential Markov Chain Monte Carlo](https://mlsanity.com/paper/2111.10210/) (2021)
- [Geometric Ergodicity in Modified Variations of Riemannian Manifold and Lagrangian Monte Carlo](https://mlsanity.com/paper/2301.01409/) (2023)
- [Repelling-Attracting Hamiltonian Monte Carlo](https://mlsanity.com/paper/2403.04607/) (2024)
- [Multi-fidelity Hamiltonian Monte Carlo](https://mlsanity.com/paper/2405.05033/) (2024)

## Abstract (arXiv)

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.