# Flow matching on homogeneous spaces

arXiv: https://arxiv.org/abs/2603.24829 · HTML: https://mlsanity.com/paper/2603.24829/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 19 references that are in the map.
- First posted on arXiv 25 Mar 2026.

## Builds on (most cited first, 17 of 17)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [Moser Flow: Divergence-based Generative Modeling on Manifolds](https://mlsanity.com/paper/2108.08052/) (2021)
- [Exponential Family Variational Flow Matching for Tabular Data Generation](https://mlsanity.com/paper/2506.05940/) (2025)
- [Controlled Generation with Equivariant Variational Flow Matching](https://mlsanity.com/paper/2506.18340/) (2025)
- [TO-FLOW: Efficient Continuous Normalizing Flows with Temporal Optimization adjoint with Moving Speed](https://mlsanity.com/paper/2203.10335/) (2022)
- [Hilbert geometry of the Siegel disk: The Siegel-Klein disk model](https://mlsanity.com/paper/2004.08160/) (2020)
- [Flow Matching on Lie Groups](https://mlsanity.com/paper/2504.00494/) (2025)

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

- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)

## Abstract (arXiv)

We propose a general framework to extend Flow Matching to homogeneous spaces, i.e. quotients of Lie groups. Our approach reformulates the problem as a flow matching task on the underlying Lie group by lifting the data distributions. This strategy avoids the potentially complicated geometry of homogeneous spaces by working directly on Lie groups, which in turn enables us reduce the problem to a Euclidean flow matching task on Lie algebras. In contrast to Riemannian Flow Matching, our method eliminates the need to define and compute premetrics or geodesics, resulting in a simpler, faster, and fully intrinsic framework.