Flow matching on homogeneous spaces

Abstract: 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.

• The paper’s main contribution is reformulating flow matching on Lie groups as flow matching on their Lie algebras to simplify computation and improve scalability.
• It introduces an intrinsic framework that lifts distributions to Lie groups, applies Lie algebra reduction, and projects back to the homogeneous space without explicit geodesic computations.
• Empirical results on spaces like the Siegel upper half-plane and S² show that leveraging Lie group structure significantly outperforms naive Euclidean approaches.

Flow Matching on Homogeneous Spaces: An Expert Analysis

Motivation and Background

Continuous Normalizing Flows (CNFs) have been a foundational technique for constructing expressive generative models through the integration of neural ODEs to transport a base noise distribution to a data distribution. Despite their theoretical appeal, simulation of ODEs in high-dimensional data regimes introduces prohibitive computational bottlenecks. The Flow Matching (FM) framework, as introduced by Lipman et al., replaces the simulation-driven training objective with a conditional, regression-based loss, sidestepping ODE integration and exhibiting improvements in scalability and efficiency.

Previous generalizations of FM to non-Euclidean structures, notably Riemannian manifolds and Lie groups, have enhanced its compatibility with geometric data but retained significant limitations, particularly the need for explicit geodesic computation or reliance on Riemannian distance metrics, which are typically intractable for general homogeneous spaces.

Main Contributions

The paper "Flow matching on homogeneous spaces" (2603.24829) presents two main methodological advances:

1. A reformulation of flow matching on Lie groups as flow matching on their Lie algebras, yielding a computationally and conceptually streamlined framework.
2. A practical extension to homogeneous spaces $G/H$ (quotients of Lie groups), leveraging lifts to the Lie group and subsequent projections to the quotient, thus bypassing the geometric complications of the homogeneous space itself.

The approach is fully intrinsic and avoids all explicit use of premetrics and geodesics. It relies on the algebraic structure of Lie groups to transport probability distributions efficiently between latent and data spaces, reducing, whenever possible, the flow matching procedure to standard Euclidean FM operating within the linear structure of the corresponding Lie algebra.

Technical Framework

The framework consists of the following essential steps:

• Lifting distributions: Given probability measures on the homogeneous space $G/H$, one utilizes global or local sections to lift the distributions to the group $G$.
• Lie algebra reduction: The standard FM loss is evaluated in the Lie algebra $\mathfrak{g}$, where elements are parameterized vectorially. The interpolation between points uses affine combinations in $\mathfrak{g}$, and the conditional vector field is identical to Euclidean FM's closed form.
• Projection: After learning the transport in $\mathfrak{g}$ and pushing it forward to $G$ via the exponential map, the results are projected back to $G/H$.

This framework is valid under the assumption of surjectivity of the exponential map (e.g., for compact/nilpotent groups), or for datasets restricted to the image of the exponential map. These conditions, while restrictive, cover many structured and physically relevant scenarios.

A key claim in the work is that Euclidean FM on $\mathfrak{g}$ with suitable left-invariant metrics is equivalent to Lie group FM with the intrinsic group norm. This correspondence facilitates implementation and encoding for high-dimensional, symmetry-structured data.

Empirical Results

Empirical validation is conducted on two representative homogeneous spaces:

• The Siegel upper half-plane $$\mathbb{H} \cong \operatorname{SL}(2, \mathbb{R}) / \operatorname{SO}(2, \mathbb{R})$$.
• The sphere $G/H$0.

Three methodological variants are compared:

1. Lie-agnostic FM: Treat data naively as residing in the ambient Euclidean space containing $G/H$1, ignoring group structure.
2. Lie group FM via Lie algebra: Employ the proposed algebraic FM formulation.
3. Parameter-encoded Lie algebra FM: Further condense Lie algebra elements into minimal parameter representations, further optimizing computational and representational efficiency.

The results demonstrate that respecting the group structure yields markedly better performance, with significant generalization gaps favoring the Lie-aware variants. While the parameter encoding does not substantially improve results in these low-dimensional toy cases, the methodology is expected to become crucial for applications where $G/H$2 is large.

Implications and Future Directions

The core implication of this work is in offering a unified, algebraic, and fully intrinsic approach to flow-based generative modeling on homogeneous spaces—removing reliance on bespoke geodesic distances or premetric structures. The reduction to well-understood Euclidean FM enables leveraging modern architectures and optimization techniques without sacrificing geometric fidelity.

For practical applications, this framework is poised to facilitate generative modeling for data intrinsically living on manifolds with significant Lie symmetry, including pose and orientation data, shape manifolds, and structured physical systems. The approach is also suggestive for equivariant and invariant model constructions, as explored in works such as "Equivariant Flow Matching" [klein2023].

Future developments could address:

• High-dimensional and sparse settings: Managing large scale homogeneous spaces or sparse data in the total space.
• Explicit $G/H$3-invariance: Incorporating or enforcing invariance under the stabilizer subgroup, crucial for quotient geometries when only classes, not representatives, are observed.
• Beyond surjective exponential maps: Relaxing the surjectivity requirement, perhaps via generalized coordinates, could expand applicability.

Conclusion

This paper introduces a robust method for conducting flow matching on homogeneous spaces by reducing it to flow matching on the Lie algebra associated with the underlying group, significantly simplifying both the theoretical and computational aspects of generative modeling for geometric data. The empirical evidence supports strong improvements over naive approaches, and the conceptual clarity potentially opens new avenues in symmetry-aware generative modeling. These contributions are likely to inform further progress in generative modeling on structured and non-Euclidean spaces.