# Convergence of Kinetic Langevin Monte Carlo on Lie groups

arXiv: https://arxiv.org/abs/2403.12012 · HTML: https://mlsanity.com/paper/2403.12012/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 37 references that are in the map.
- First posted on arXiv 18 Mar 2024.

## Builds on (most cited first, 11 of 11)

- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev](https://mlsanity.com/paper/2010.05263/) (2020)
- [Towards Riemannian Accelerated Gradient Methods](https://mlsanity.com/paper/1806.02812/) (2018)
- [Variational Optimization on Lie Groups, with Examples of Leading (Generalized) Eigenvalue Problems](https://mlsanity.com/paper/2001.10006/) (2020)
- [Momentum Stiefel Optimizer, with Applications to Suitably-Orthogonal Attention, and Optimal Transport](https://mlsanity.com/paper/2205.14173/) (2022)
- [Efficient Sampling on Riemannian Manifolds via Langevin MCMC](https://mlsanity.com/paper/2402.10357/) (2024)

## Built on by (most cited first, 6 of 6)

- [Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups](https://mlsanity.com/paper/2405.16381/) (2024)
- [Kinetic Langevin Diffusion for Crystalline Materials Generation](https://mlsanity.com/paper/2507.03602/) (2025)
- [Quantum State Generation with Structure-Preserving Diffusion Model](https://mlsanity.com/paper/2404.06336/) (2024)
- [Diffusion Generative Modeling on Lie Group Representations](https://mlsanity.com/paper/2502.02513/) (2025)
- [Inverting Data Transformations via Diffusion Sampling](https://mlsanity.com/paper/2602.08267/) (2026)
- [Quantitative Convergences of Lie Group Momentum Optimizers](https://mlsanity.com/paper/2405.20390/) (2024)

## Abstract (arXiv)

Explicit, momentum-based dynamics for optimizing functions defined on Lie groups was recently constructed, based on techniques such as variational optimization and left trivialization. We appropriately add tractable noise to the optimization dynamics to turn it into a sampling dynamics, leveraging the advantageous feature that the trivialized momentum variable is Euclidean despite that the potential function lives on a manifold. We then propose a Lie-group MCMC sampler, by delicately discretizing the resulting kinetic-Langevin-type sampling dynamics. The Lie group structure is exactly preserved by this discretization. Exponential convergence with explicit convergence rate for both the continuous dynamics and the discrete sampler are then proved under W2 distance. Only compactness of the Lie group and geodesically L-smoothness of the potential function are needed. To the best of our knowledge, this is the first convergence result for kinetic Langevin on curved spaces, and also the first quantitative result that requires no convexity or, at least not explicitly, any common relaxation such as isoperimetry.