# Geometric integrators and the Hamiltonian Monte Carlo method

arXiv: https://arxiv.org/abs/1711.05337 · HTML: https://mlsanity.com/paper/1711.05337/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Nov 2017.

## Built on by (most cited first, 25 of 25)

- [Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.00887/) (2021)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [Efficient Learning of the Parameters of Non-Linear Models using Differentiable Resampling in Particle Filters](https://mlsanity.com/paper/2111.01409/) (2021)
- [GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2404.15253/) (2024)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration](https://mlsanity.com/paper/2211.11003/) (2022)
- [Hamiltonian Score Matching and Generative Flows](https://mlsanity.com/paper/2410.20470/) (2024)
- [Second Order Ensemble Langevin Method for Sampling and Inverse Problems](https://mlsanity.com/paper/2208.04506/) (2022)
- [Entropy-based adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2110.14625/) (2021)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [A Unifying and Canonical Description of Measure-Preserving Diffusions](https://mlsanity.com/paper/2105.02845/) (2021)
- [Nonparametric Hamiltonian Monte Carlo](https://mlsanity.com/paper/2106.10238/) (2021)
- [Better Training using Weight-Constrained Stochastic Dynamics](https://mlsanity.com/paper/2106.10704/) (2021)
- [On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers](https://mlsanity.com/paper/2307.03460/) (2023)
- [Tuning diagonal scale matrices for HMC](https://mlsanity.com/paper/2403.07495/) (2024)
- [Log-density gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods](https://mlsanity.com/paper/2211.01746/) (2022)
- [Randomised Splitting Methods and Stochastic Gradient Descent](https://mlsanity.com/paper/2504.04274/) (2025)
- [The Within-Orbit Adaptive Leapfrog No-U-Turn Sampler](https://mlsanity.com/paper/2506.18746/) (2025)
- [Randomized Runge-Kutta-Nystr"om Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo](https://mlsanity.com/paper/2310.07399/) (2023)
- [A Theoretical Comparison of No-U-Turn Sampler Variants: Necessary and Sufficient Convergence Conditions and Mixing Time Analysis under Gaussian Targets](https://mlsanity.com/paper/2603.18640/) (2026)
- [Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo](https://mlsanity.com/paper/2102.13068/) (2021)
- [Learning variational autoencoders via MCMC speed measures](https://mlsanity.com/paper/2308.13731/) (2023)
- [Numerical Generalized Randomized Hamiltonian Monte Carlo for piecewise smooth target densities](https://mlsanity.com/paper/2504.18210/) (2025)
- [Accelerated sampling using SamAdams variable timesteps and position-adaptive Langevin dynamics](https://mlsanity.com/paper/2606.26881/) (2026)
- [Optimal scaling of MCMC algorithms: the Hamiltonian approach](https://mlsanity.com/paper/2607.00586/) (2026)

## Abstract (arXiv)

This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications on the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behavior of HMC as the dimensionality of the target distribution increases.