# Flow Matching on Lie Groups

arXiv: https://arxiv.org/abs/2504.00494 · HTML: https://mlsanity.com/paper/2504.00494/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 5 of its 10 references that are in the map.
- First posted on arXiv 1 Apr 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 5 of 5)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Grounding Continuous Representations in Geometry: Equivariant Neural Fields](https://mlsanity.com/paper/2406.05753/) (2024)

## Built on by (most cited first, 4 of 4)

- [Discovering Symmetry Groups with Flow Matching](https://mlsanity.com/paper/2512.20043/) (2025)
- [Flow matching on homogeneous spaces](https://mlsanity.com/paper/2603.24829/) (2026)
- [Cartan flow matching](https://mlsanity.com/paper/2605.03588/) (2026)
- [Equivariant Latent Alignment via Flow Matching under Group Symmetries](https://mlsanity.com/paper/2605.30705/) (2026)

## Abstract (arXiv)

Flow Matching (FM) is a recent generative modelling technique: we aim to learn how to sample from distribution mathfrakX1 by flowing samples from some distribution mathfrakX0 that is easy to sample from. The key trick is that this flow field can be trained while conditioning on the end point in mathfrakX1: given an end point, simply move along a straight line segment to the end point (Lipman et al. 2022). However, straight line segments are only well-defined on Euclidean space. Consequently, Chen and Lipman (2023) generalised the method to FM on Riemannian manifolds, replacing line segments with geodesics or their spectral approximations. We take an alternative point of view: we generalise to FM on Lie groups by instead substituting exponential curves for line segments. This leads to a simple, intrinsic, and fast implementation for many matrix Lie groups, since the required Lie group operations (products, inverses, exponentials, logarithms) are simply given by the corresponding matrix operations. FM on Lie groups could then be used for generative modelling with data consisting of sets of features (in ℝn) and poses (in some Lie group), e.g. the latent codes of Equivariant Neural Fields (Wessels et al. 2025).