# Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation

arXiv: https://arxiv.org/abs/1710.06261 · HTML: https://mlsanity.com/paper/1710.06261/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2017.

## Built on by (most cited first, 30 of 32)

- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.00887/) (2021)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [Sampling in Constrained Domains with Orthogonal-Space Variational Gradient Descent](https://mlsanity.com/paper/2210.06447/) (2022)
- [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023)
- [Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators](https://mlsanity.com/paper/2210.07219/) (2022)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [An Introduction to Hamiltonian Monte Carlo Method for Sampling](https://mlsanity.com/paper/2108.12107/) (2021)
- [Sampling from Log-Concave Distributions with Infinity-Distance Guarantees](https://mlsanity.com/paper/2111.04089/) (2021)
- [Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithms](https://mlsanity.com/paper/2206.03792/) (2022)
- [Sampling with Barriers: Faster Mixing via Lewis Weights](https://mlsanity.com/paper/2303.00480/) (2023)
- [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024)
- [Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo](https://mlsanity.com/paper/2210.11925/) (2022)
- [Efficient Sampling on Riemannian Manifolds via Langevin MCMC](https://mlsanity.com/paper/2402.10357/) (2024)
- [The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models](https://mlsanity.com/paper/2405.17068/) (2024)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [New affine invariant ensemble samplers and their dimensional scaling](https://mlsanity.com/paper/2505.02987/) (2025)
- [Efficient computation of the volume of a polytope in high-dimensions using Piecewise Deterministic Markov Processes](https://mlsanity.com/paper/2202.09129/) (2022)
- [Learning Rate Free Sampling in Constrained Domains](https://mlsanity.com/paper/2305.14943/) (2023)
- [R'enyi-infinity constrained sampling with d3 membership queries](https://mlsanity.com/paper/2407.12967/) (2024)
- [Anchored Langevin Algorithms](https://mlsanity.com/paper/2509.19455/) (2025)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)
- [Tukey Depth Mechanisms for Practical Private Mean Estimation](https://mlsanity.com/paper/2502.18698/) (2025)
- [Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walk](https://mlsanity.com/paper/2206.09384/) (2022)

## Abstract (arXiv)

We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the method with the manifold defined by the log barrier function to the problems of (1) uniformly sampling a polytope and (2) computing its volume, the latter by extending Gaussian cooling to the manifold setting. In both cases, the total number of steps needed is O*(mn^frac23), improving the state of the art. A key ingredient of our analysis is a proof of an analog of the KLS conjecture for Gibbs distributions over manifolds.