# Time-Inhomogeneous Preconditioned Langevin Dynamics

arXiv: https://arxiv.org/abs/2605.06091 · HTML: https://mlsanity.com/paper/2605.06091/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 41 references that are in the map.
- First posted on arXiv 7 May 2026.

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

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Exponential ergodicity of mirror-Langevin diffusions](https://mlsanity.com/paper/2005.09669/) (2020)
- [The Tamed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1710.05559/) (2017)
- [Langevin diffusions and the Metropolis-adjusted Langevin algorithm](https://mlsanity.com/paper/1309.2983/) (2013)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Ensemble preconditioning for Markov chain Monte Carlo simulation](https://mlsanity.com/paper/1607.03954/) (2016)
- [Stochastic Quasi-Newton Langevin Monte Carlo](https://mlsanity.com/paper/1602.03442/) (2016)
- [Optimal Preconditioning and Fisher Adaptive Langevin Sampling](https://mlsanity.com/paper/2305.14442/) (2023)
- [Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.12499/) (2022)
- [Non-asymptotic Analysis of Diffusion Annealed Langevin Monte Carlo for Generative Modelling](https://mlsanity.com/paper/2502.09306/) (2025)
- [Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics](https://mlsanity.com/paper/2410.09697/) (2024)
- [Computed Tomography Reconstruction using Generative Energy-Based Priors](https://mlsanity.com/paper/2203.12658/) (2022)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [Sampling and estimation on manifolds using the Langevin diffusion](https://mlsanity.com/paper/2312.14882/) (2023)
- [High-accuracy sampling from constrained spaces with the Metropolis-adjusted Preconditioned Langevin Algorithm](https://mlsanity.com/paper/2412.18701/) (2024)
- [Energy-based models for inverse imaging problems](https://mlsanity.com/paper/2507.12432/) (2025)
- [Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics](https://mlsanity.com/paper/2602.01449/) (2026)
- [Forward-KL Convergence of Time-Inhomogeneous Langevin Diffusions](https://mlsanity.com/paper/2601.22349/) (2026)
- [Riemannian Langevin Dynamics: Strong Convergence of Geometric Euler-Maruyama Scheme](https://mlsanity.com/paper/2603.03626/) (2026)

## Abstract (arXiv)

Langevin sampling from distributions of the form p(x) propto exp(-Psi(x)) faces two major challenges: (global) mode coverage and (local) mode exploration. The first challenge is particularly relevant for multi-modal distributions with disjoint modes, whereas the second arises when the potential Psi exhibits diverse and ill-conditioned local mode geometry. To address these challenges, a common approach is to precondition Langevin dynamics with problem-specific information, such as the sample covariance or the local curvature of Psi. However, existing preconditioner choices inherently involve a trade-off between global mode coverage and local mode exploration, and no prior method resolves both simultaneously. To overcome this limitation, we propose the TIPreL, which introduces a time- and position-dependent preconditioner. This design effectively addresses both challenges mentioned above within a single framework. We establish convergence of the resulting dynamics in the Wasserstein-2 distance both in continuous time and for a tamed Euler discretization. In particular, our analysis extends the existing state of the art by proving convergence under time- and space-dependent diffusion coefficients, and only locally Lipschitz drifts, which has not been covered by prior work. Finally, we experimentally compare TIPreL with competing preconditioning schemes on a two-dimensional, severely ill-posed example and on a Bayesian logistic regression task in higher dimensions, confirming the efficiency of the proposed method.