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Flow Matching on Lie Groups

Finn M. Sherry, Bart M. N. Smets

TL;DR

This paper addresses generative modelling on manifolds by extending Flow Matching (FM) from Euclidean space to Lie groups. It proposes a Lie-group FM that uses exponential-curve integral trajectories with a conditional vector field, enabling simulation-free, intrinsic interpolation on groups with surjective exponential maps. The authors show that Euclidean FM is a special case under the translation group and demonstrate implementations on $\mathrm{SE}(2)$, $\mathrm{SO}(3)$, and $\mathrm{SE}(2)\times\mathbb{R}^2$ using simple neural networks and standard matrix operations. This approach opens up new possibilities for geometrically structured data, such as latent codes of Equivariant Neural Fields, providing a principled framework for generative modelling on manifolds.

Abstract

Flow Matching (FM) is a recent generative modelling technique: we aim to learn how to sample from distribution $\mathfrak{X}_1$ by flowing samples from some distribution $\mathfrak{X}_0$ that is easy to sample from. The key trick is that this flow field can be trained while conditioning on the end point in $\mathfrak{X}_1$: 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 $\mathbb{R}^n$) and poses (in some Lie group), e.g. the latent codes of Equivariant Neural Fields (Wessels et al. 2025).

Flow Matching on Lie Groups

TL;DR

This paper addresses generative modelling on manifolds by extending Flow Matching (FM) from Euclidean space to Lie groups. It proposes a Lie-group FM that uses exponential-curve integral trajectories with a conditional vector field, enabling simulation-free, intrinsic interpolation on groups with surjective exponential maps. The authors show that Euclidean FM is a special case under the translation group and demonstrate implementations on , , and using simple neural networks and standard matrix operations. This approach opens up new possibilities for geometrically structured data, such as latent codes of Equivariant Neural Fields, providing a principled framework for generative modelling on manifolds.

Abstract

Flow Matching (FM) is a recent generative modelling technique: we aim to learn how to sample from distribution by flowing samples from some distribution that is easy to sample from. The key trick is that this flow field can be trained while conditioning on the end point in : 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 ) and poses (in some Lie group), e.g. the latent codes of Equivariant Neural Fields (Wessels et al. 2025).

Paper Structure

This paper contains 13 sections, 2 theorems, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Euclidean Flow Matching.
  3. Riemannian Flow Matching.
  4. Our Contribution.
  5. Lie Group Flow Matching
  6. Reconsidering Euclidean Flow Matching.
  7. Flow Matching on SE(2).
  8. Flow Matching on Matrix Groups.
  9. Flow Matching on Product Groups.
  10. Flow Matching on Homogeneous Spaces.
  11. Experiments
  12. Conclusion & Future Work.
  13. Acknowledgements.

Key Result

proposition 1

The integral curves of the vector field $u_t(\cdot \mid g_1): [0, 1] \to \mathop{\mathrm{\Gamma}}\nolimits(TG)$, with $g_1 \in G$, given by are the exponential curves ending in $g_1$.

Figures (3)

  • Figure 1: FM on $\mathop{\mathrm{SE}}\nolimits(2)$, interpreted as the space of planar positions and orientations. Top: flowing from horizontal line to vertical line. Bottom: flowing from vertical line to circle.
  • Figure 2: FM on $\mathop{\mathrm{SO}}\nolimits(3)$, interpreted as the space of spherical positions and orientations. Top: flowing from horizontal line to vertical line. Bottom: flowing from vertical line to circle.
  • Figure 3: FM on $\mathop{\mathrm{SE}}\nolimits(2) \times \mathbb{R}^2$, with $\mathop{\mathrm{SE}}\nolimits(2)$ interpreted as the space of planar positions and orientations. Note that this shows the flow of a single pair of distributions $\mathfrak{X}_0$ and $\mathfrak{X}_1$: the top row shows the $\mathop{\mathrm{SE}}\nolimits(2)$ component and bottom row shows the $\mathbb{R}^2$ component.

Theorems & Definitions (9)

  • definition 1: Lie Group Operations
  • remark 1
  • proposition 1: Lie Group Flow Field
  • proof
  • remark 2
  • theorem 1: Optimise on Conditional Loss
  • proof
  • definition 2: Special Euclidean Group
  • definition 3: Special Orthogonal Group