# Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/2210.11925 · HTML: https://mlsanity.com/paper/2210.11925/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 58 references that are in the map.
- First posted on arXiv 21 Oct 2022.

## Builds on (most cited first, 10 of 10)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017)
- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Geodesic Walks in Polytopes](https://mlsanity.com/paper/1606.04696/) (2016)
- [Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing](https://mlsanity.com/paper/1612.01507/) (2016)
- [Introducing an Explicit Symplectic Integration Scheme for Riemannian Manifold Hamiltonian Monte Carlo](https://mlsanity.com/paper/1910.06243/) (2019)
- [Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators](https://mlsanity.com/paper/2210.07219/) (2022)
- [Sampling constrained probability distributions using Spherical Augmentation](https://mlsanity.com/paper/1506.05936/) (2015)

## Built on by (most cited first, 6 of 6)

- [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023)
- [Constrained Sampling with Primal-Dual Langevin Monte Carlo](https://mlsanity.com/paper/2411.00568/) (2024)
- [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024)
- [Sampling with Shielded Langevin Monte Carlo Using Navigation Potentials](https://mlsanity.com/paper/2512.22153/) (2025)
- [Sampling from multi-modal distributions on Riemannian manifolds with training-free stochastic interpolants](https://mlsanity.com/paper/2602.00641/) (2026)
- [Kinetic Langevin Splitting Schemes for Constrained Sampling](https://mlsanity.com/paper/2603.23397/) (2026)

## Abstract (arXiv)

In this paper, we propose Barrier Hamiltonian Monte Carlo (BHMC), a version of the HMC algorithm which aims at sampling from a Gibbs distribution π on a manifold M, endowed with a Hessian metric mathfrakg derived from a self-concordant barrier. Our method relies on Hamiltonian dynamics which comprises mathfrakg. Therefore, it incorporates the constraints defining M and is able to exploit its underlying geometry. However, the corresponding Hamiltonian dynamics is defined via non separable Ordinary Differential Equations (ODEs) in contrast to the Euclidean case. It implies unavoidable bias in existing generalization of HMC to Riemannian manifolds. In this paper, we propose a new filter step, called "involution checking step", to address this problem. This step is implemented in two versions of BHMC, coined continuous BHMC (c-BHMC) and numerical BHMC (n-BHMC) respectively. Our main results establish that these two new algorithms generate reversible Markov chains with respect to π and do not suffer from any bias in comparison to previous implementations. Our conclusions are supported by numerical experiments where we consider target distributions defined on polytopes.