# Langevin diffusions and the Metropolis-adjusted Langevin algorithm

arXiv: https://arxiv.org/abs/1309.2983 · HTML: https://mlsanity.com/paper/1309.2983/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2013.

## Built on by (most cited first, 29 of 29)

- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [A Comprehensive Survey on Knowledge Distillation of Diffusion Models](https://mlsanity.com/paper/2304.04262/) (2023)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
- [Optimal Preconditioning and Fisher Adaptive Langevin Sampling](https://mlsanity.com/paper/2305.14442/) (2023)
- [Entropy-based Training Methods for Scalable Neural Implicit Sampler](https://mlsanity.com/paper/2306.04952/) (2023)
- [Lagrangian Manifold Monte Carlo on Monge Patches](https://mlsanity.com/paper/2202.00755/) (2022)
- [Normalizing flow sampling with Langevin dynamics in the latent space](https://mlsanity.com/paper/2305.12149/) (2023)
- [Kinetic Interacting Particle Langevin Monte Carlo](https://mlsanity.com/paper/2407.05790/) (2024)
- [Faster Sampling via Stochastic Gradient Proximal Sampler](https://mlsanity.com/paper/2405.16734/) (2024)
- [A Unifying and Canonical Description of Measure-Preserving Diffusions](https://mlsanity.com/paper/2105.02845/) (2021)
- [Geometry-informed irreversible perturbations for accelerated convergence of Langevin dynamics](https://mlsanity.com/paper/2108.08247/) (2021)
- [ELF: Federated Langevin Algorithms with Primal, Dual and Bidirectional Compression](https://mlsanity.com/paper/2303.04622/) (2023)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Warped geometric information on the optimisation of Euclidean functions](https://mlsanity.com/paper/2308.08305/) (2023)
- [AdamMCMC: Combining Metropolis Adjusted Langevin with Momentum-based Optimization](https://mlsanity.com/paper/2312.14027/) (2023)
- [Transport map unadjusted Langevin algorithms: learning and discretizing perturbed samplers](https://mlsanity.com/paper/2302.07227/) (2023)
- [The Application of Zig-Zag Sampler in Sequential Markov Chain Monte Carlo](https://mlsanity.com/paper/2111.10210/) (2021)
- [Clarifying MCMC-based training of modern EBMs : Contrastive Divergence versus Maximum Likelihood](https://mlsanity.com/paper/2202.12176/) (2022)
- [Geometric Ergodicity in Modified Variations of Riemannian Manifold and Lagrangian Monte Carlo](https://mlsanity.com/paper/2301.01409/) (2023)
- [An Improved Analysis of Langevin Algorithms with Prior Diffusion for Non-Log-Concave Sampling](https://mlsanity.com/paper/2403.06183/) (2024)
- [Robust Inference of Dynamic Covariance Using Wishart Processes and Sequential Monte Carlo](https://mlsanity.com/paper/2406.04796/) (2024)
- [Feynman-Kac Operator Expectation Estimator](https://mlsanity.com/paper/2407.02010/) (2024)
- [Denoising Fisher Training For Neural Implicit Samplers](https://mlsanity.com/paper/2411.01453/) (2024)
- [Backpropagation-Free Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2505.18081/) (2025)
- [Sampling on Metric Graphs](https://mlsanity.com/paper/2512.02175/) (2025)
- [Time-Inhomogeneous Preconditioned Langevin Dynamics](https://mlsanity.com/paper/2605.06091/) (2026)
- [Gradient-free Riemannian Langevin Sampler](https://mlsanity.com/paper/2607.07519/) (2026)
- [On skew-symmetric distributions and their use in Monte Carlo sampling algorithms: coordinate-free, Gibbs-style and manifold versions of the Barker proposal](https://mlsanity.com/paper/2610.01448/) (2026)

## Abstract (arXiv)

We provide a clarification of the description of Langevin diffusions on Riemannian manifolds and of the measure underlying the invariant density. As a result we propose a new position-dependent Metropolis-adjusted Langevin algorithm (MALA) based upon a Langevin diffusion in ℝd which has the required invariant density with respect to Lebesgue measure. We show that our diffusion and the diffusion upon which a previously-proposed position-dependent MALA is based are equivalent in some cases but are distinct in general. A simulation study illustrates the gain in efficiency provided by the new position-dependent MALA.