mlsanityData to 14 Oct 2026

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

arXiv 1710.06261 ↗First posted 17 Oct 2017cs.DS math.FA stat.MLcited by 32

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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.

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

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Citing papers per year: 2021: 6; 2022: 9; 2023: 7; 2024: 6; 2025: 4.

32 papers.

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  1. 2023Diffusion Models for Constrained Domainscited by 63
  2. 2022Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Spacecited by 28
  3. 2023Metropolis Sampling for Constrained Diffusion Modelscited by 28
  4. 2021A Proximal Algorithm for Sampling from Non-smooth Potentialscited by 24
  5. 2022A Proximal Algorithm for Samplingcited by 22
  6. 2022Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Samplingcited by 22
  7. 2024In-and-Out: Algorithmic Diffusion for Sampling Convex Bodiescited by 19
  8. 2021Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlocited by 18
  9. 2021Provable Compressed Sensing with Generative Priors via Langevin Dynamicscited by 14
  10. 2022Sampling in Constrained Domains with Orthogonal-Space Variational Gradient Descentcited by 11
  11. 2023Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithmcited by 11
  12. 2022Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integratorscited by 10
20 more
  1. 2023Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Samplercited by 10
  2. 2021An Introduction to Hamiltonian Monte Carlo Method for Samplingcited by 8
  3. 2021Sampling from Log-Concave Distributions with Infinity-Distance Guaranteescited by 8
  4. 2022Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithmscited by 7
  5. 2023Sampling with Barriers: Faster Mixing via Lewis Weightscited by 7
  6. 2024Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamicscited by 7
  7. 2022Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlocited by 6
  8. 2024Efficient Sampling on Riemannian Manifolds via Langevin MCMCcited by 6
  9. 2024The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Modelscited by 6
  10. 2023Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithmscited by 4
  11. 2025New affine invariant ensemble samplers and their dimensional scalingcited by 4
  12. 2022Efficient computation of the volume of a polytope in high-dimensions using Piecewise Deterministic Markov Processescited by 3
  13. 2023Learning Rate Free Sampling in Constrained Domainscited by 3
  14. 2024R'enyi-infinity constrained sampling with d3 membership queriescited by 3
  15. 2025Anchored Langevin Algorithmscited by 3
  16. 2025Oracle-based Uniform Sampling from Convex Bodiescited by 3
  17. 2025Tukey Depth Mechanisms for Practical Private Mean Estimationcited by 2
  18. 2022Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walkcited by 1
  19. 2024Faster Sampling from Log-Concave Densities over Polytopes via Efficient Linear Solverscited by 1
  20. 2021Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlocited by 0