# Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

arXiv: https://arxiv.org/abs/1705.08964 · HTML: https://mlsanity.com/paper/1705.08964/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2017.

## Built on by (most cited first, 30 of 43)

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- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [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)
- [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited](https://mlsanity.com/paper/2306.08494/) (2023)
- [Sampling from Log-Concave Distributions with Infinity-Distance Guarantees](https://mlsanity.com/paper/2111.04089/) (2021)
- [Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes](https://mlsanity.com/paper/2207.04387/) (2022)
- [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024)
- [Non-Reversible Langevin Algorithms for Constrained Sampling](https://mlsanity.com/paper/2501.11743/) (2025)
- [Constrained Langevin Algorithms with L-mixing External Random Variables](https://mlsanity.com/paper/2205.14192/) (2022)
- [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025)
- [Sensing Cox Processes via Posterior Sampling and Positive Bases](https://mlsanity.com/paper/2110.11181/) (2021)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions](https://mlsanity.com/paper/2502.03458/) (2025)
- [Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis](https://mlsanity.com/paper/2503.19066/) (2025)
- [Convergence Error Analysis of Reflected Gradient Langevin Dynamics for Globally Optimizing Non-Convex Constrained Problems](https://mlsanity.com/paper/2203.10215/) (2022)
- [R'enyi-infinity constrained sampling with d3 membership queries](https://mlsanity.com/paper/2407.12967/) (2024)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)
- [PSI3D: Plug-and-Play 3D Stochastic Inference with Slice-wise Latent Diffusion Prior](https://mlsanity.com/paper/2512.18367/) (2025)
- [Randomized Midpoint Method for Log-Concave Sampling under Constraints](https://mlsanity.com/paper/2405.15379/) (2024)
- [Accelerating Constrained Sampling: A Large Deviations Approach](https://mlsanity.com/paper/2506.07816/) (2025)
- [Decentralized Proximal Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2605.00723/) (2026)
- [Improved Guarantees for Langevin Monte Carlo with Average Smoothness](https://mlsanity.com/paper/2605.31413/) (2026)
- [Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling](https://mlsanity.com/paper/2608.13467/) (2026)
- [Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walk](https://mlsanity.com/paper/2206.09384/) (2022)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)

## Abstract (arXiv)

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body mathsfK. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with mathsfK. Explicit convergence bounds in total variation norm and in Wasserstein distance of order 1 are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.