# Transport map unadjusted Langevin algorithms: learning and discretizing perturbed samplers

arXiv: https://arxiv.org/abs/2302.07227 · HTML: https://mlsanity.com/paper/2302.07227/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 61 references that are in the map.
- First posted on arXiv 14 Feb 2023.

## Builds on (most cited first, 23 of 23)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Exponential ergodicity of mirror-Langevin diffusions](https://mlsanity.com/paper/2005.09669/) (2020)
- [Langevin diffusions and the Metropolis-adjusted Langevin algorithm](https://mlsanity.com/paper/1309.2983/) (2013)
- [A transport-based multifidelity preconditioner for Markov chain Monte Carlo](https://mlsanity.com/paper/1808.09379/) (2018)
- [Improving the convergence of reversible samplers](https://mlsanity.com/paper/1601.08118/) (2016)
- [Information-geometric Markov Chain Monte Carlo methods using Diffusions](https://mlsanity.com/paper/1403.7957/) (2014)
- [Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/2201.08349/) (2022)
- [Geometry-informed irreversible perturbations for accelerated convergence of Langevin dynamics](https://mlsanity.com/paper/2108.08247/) (2021)
- [Optimal Scaling of the MALA algorithm with Irreversible Proposals for Gaussian targets](https://mlsanity.com/paper/1702.01777/) (2017)

## Built on by (most cited first, 1 of 1)

- [Posterior sampling by source-space MCMC via prior-based few-step transport maps](https://mlsanity.com/paper/2610.01034/) (2026)

## Abstract (arXiv)

Langevin dynamics are widely used in sampling high-dimensional, non-Gaussian distributions whose densities are known up to a normalizing constant. In particular, there is strong interest in unadjusted Langevin algorithms (ULA), which directly discretize Langevin dynamics to estimate expectations over the target distribution. We study the use of transport maps that approximately normalize a target distribution as a way to precondition and accelerate the convergence of Langevin dynamics. We show that in continuous time, when a transport map is applied to Langevin dynamics, the result is a Riemannian manifold Langevin dynamics (RMLD) with metric defined by the transport map. We also show that applying a transport map to an irreversibly-perturbed ULA results in a geometry-informed irreversible perturbation (GiIrr) of the original dynamics. These connections suggest more systematic ways of learning metrics and perturbations, and also yield alternative discretizations of the RMLD described by the map, which we study. Under appropriate conditions, these discretized processes can be endowed with non-asymptotic bounds describing convergence to the target distribution in 2-Wasserstein distance. Illustrative numerical results complement our theoretical claims.