# Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space

arXiv: https://arxiv.org/abs/2202.01908 · HTML: https://mlsanity.com/paper/2202.01908/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 58 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 15 of 15)

- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Exponential ergodicity of mirror-Langevin diffusions](https://mlsanity.com/paper/2005.09669/) (2020)
- [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017)
- [Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo](https://mlsanity.com/paper/2002.04121/) (2020)
- [Stochastic Quasi-Newton Langevin Monte Carlo](https://mlsanity.com/paper/1602.03442/) (2016)
- [Geodesic Walks in Polytopes](https://mlsanity.com/paper/1606.04696/) (2016)
- [Introducing an Explicit Symplectic Integration Scheme for Riemannian Manifold Hamiltonian Monte Carlo](https://mlsanity.com/paper/1910.06243/) (2019)
- [Convergence of Gibbs Sampling: Coordinate Hit-and-Run Mixes Fast](https://mlsanity.com/paper/2009.11338/) (2020)
- [Universal Barrier is n-Self-Concordant](https://mlsanity.com/paper/1809.03011/) (2018)
- [Volesti: Volume Approximation and Sampling for Convex Polytopes in R](https://mlsanity.com/paper/2007.01578/) (2020)
- [On the mixing time of coordinate Hit-and-Run](https://mlsanity.com/paper/2009.14004/) (2020)

## Built on by (most cited first, 28 of 28)

- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [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)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Sampling and Integration of Logconcave Functions by Algorithmic Diffusion](https://mlsanity.com/paper/2411.13462/) (2024)
- [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)
- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [Constrained Diffusers for Safe Planning and Control](https://mlsanity.com/paper/2506.12544/) (2025)
- [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)
- [Faster logconcave sampling from a cold start in high dimension](https://mlsanity.com/paper/2505.01937/) (2025)
- [Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo](https://mlsanity.com/paper/2210.11925/) (2022)
- [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)
- [R'enyi-infinity constrained sampling with d3 membership queries](https://mlsanity.com/paper/2407.12967/) (2024)
- [Randomized Midpoint Method for Log-Concave Sampling under Constraints](https://mlsanity.com/paper/2405.15379/) (2024)
- [Beyond the d2.5-mixing bound for Dikin walks on polytopes](https://mlsanity.com/paper/2607.13943/) (2026)
- [PolytopeWalk: Sparse MCMC Sampling over Polytopes](https://mlsanity.com/paper/2412.06629/) (2024)
- [Manifold Sampling via Entropy Maximization](https://mlsanity.com/paper/2605.12338/) (2026)
- [Uncertainty-Aware Gaussian Map for Vision-Language Navigation](https://mlsanity.com/paper/2605.26503/) (2026)
- [Sampling Ex-Post Group-Fair Rankings](https://mlsanity.com/paper/2203.00887/) (2022)
- [Sampling Constrained Continuous Probability Distributions: A Review](https://mlsanity.com/paper/2209.12403/) (2022)
- [Log-Concave Coupling for Sampling Neural Net Posteriors](https://mlsanity.com/paper/2407.18802/) (2024)
- [Sampling with Shielded Langevin Monte Carlo Using Navigation Potentials](https://mlsanity.com/paper/2512.22153/) (2025)
- [Kinetic Langevin Splitting Schemes for Constrained Sampling](https://mlsanity.com/paper/2603.23397/) (2026)

## Abstract (arXiv)

We demonstrate for the first time that ill-conditioned, non-smooth, constrained distributions in very high dimension, upwards of 100,000, can be sampled efficiently textitin practice. Our algorithm incorporates constraints into the Riemannian version of Hamiltonian Monte Carlo and maintains sparsity. This allows us to achieve a mixing rate independent of smoothness and condition numbers. On benchmark data sets in systems biology and linear programming, our algorithm outperforms existing packages by orders of magnitude. In particular, we achieve a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D). Our package has been incorporated into the COBRA toolbox.