# Introducing an Explicit Symplectic Integration Scheme for Riemannian Manifold Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/1910.06243 · HTML: https://mlsanity.com/paper/1910.06243/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2019.

## Built on by (most cited first, 10 of 10)

- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Lagrangian Manifold Monte Carlo on Monge Patches](https://mlsanity.com/paper/2202.00755/) (2022)
- [Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators](https://mlsanity.com/paper/2210.07219/) (2022)
- [Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo](https://mlsanity.com/paper/2210.11925/) (2022)
- [Exponential Family Model-Based Reinforcement Learning via Score Matching](https://mlsanity.com/paper/2112.14195/) (2021)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [Direct Amortized Likelihood Ratio Estimation](https://mlsanity.com/paper/2311.10571/) (2023)
- [Semi-Empirical Objective Functions for MCMC Proposal Optimization](https://mlsanity.com/paper/2106.02104/) (2021)
- [Antithetic Riemannian Manifold And Quantum-Inspired Hamiltonian Monte Carlo](https://mlsanity.com/paper/2107.02070/) (2021)
- [Fast Riemannian-manifold Hamiltonian Monte Carlo for hierarchical Gaussian-process models](https://mlsanity.com/paper/2511.06407/) (2025)

## Abstract (arXiv)

We introduce a recent symplectic integration scheme derived for solving physically motivated systems with non-separable Hamiltonians. We show its relevance to Riemannian manifold Hamiltonian Monte Carlo (RMHMC) and provide an alternative to the currently used generalised leapfrog symplectic integrator, which relies on solving multiple fixed point iterations to convergence. Via this approach, we are able to reduce the number of higher-order derivative calculations per leapfrog step. We explore the implications of this integrator and demonstrate its efficacy in reducing the computational burden of RMHMC. Our code is provided in a new open-source Python package, hamiltorch.