Flow Matching on Lie Groups

• The paper introduces an intrinsic framework that extends flow matching from Euclidean spaces to Lie groups by leveraging the exponential map and Lie algebra for interpolation.
• It unifies conditional and continuous-time flow matching with Lie-theoretic machine learning to enable symmetry discovery and equivariant generative modeling.
• Experimental results validate the method’s efficacy in recovering group symmetries and enhancing mode discrimination in both synthetic and realistic datasets.

Flow matching on Lie groups is a generative modeling and symmetry discovery technique that exploits the geometry and group structure of matrix Lie groups to formulate flows and distributions intrinsically on these manifolds. The approach generalizes flow matching from Euclidean spaces, where linear interpolation is well-defined, to Lie groups, where interpolation is performed along exponential curves governed by the Lie algebra. This methodology unifies recent advances in conditional and continuous-time flow matching, Lie-theoretic machine learning, symmetry discovery, and equivariant generative modeling.

1. Foundational Principles of Flow Matching on Lie Groups

Classic flow matching seeks to transport samples from a source distribution $p_0$ (typically simple, such as a Gaussian or uniform) to a target distribution $p_1$ along a path parameterized by $t \in [0,1]$, using a vector field $v_t$. In Euclidean settings, straight-line interpolation and associated velocity fields are used, which do not generally extend to manifolds or group-valued data. For Lie groups $G$, the left-invariant geometry and the exponential map from the Lie algebra $\mathfrak{g}$ facilitate an intrinsic alternative. Exponential curves of the form

$$g_t = g_0 \exp\left( t \, \log(g_0^{-1} g_1) \right)$$

interpolate between $g_0$ and $g_1$ in $G$, under the assumption that $g_0^{-1} g_1$ is in the image of the exponential map. The velocity field inducing this curve at time $t$ is given by

$$\frac{d}{dt} g_t = g_0 \exp\left(t \log(g_0^{-1} g_1)\right) \log(g_0^{-1} g_1) = (L_{g_t})_* \log(g_0^{-1} g_1)$$

where $(L_g)_*$ denotes the differential of left-multiplication by $g$. This formulation is intrinsic to the group and does not depend on extrinsic coordinate representations (Sherry et al., 1 Apr 2025).

The loss for flow matching on Lie groups is framed as

$$L_{LieCFM}(\theta) = \mathbb{E}_{t, g_0, g_1}\, \left\| v_t^\theta(\tilde{g}_t, t, g_1) - \log(\tilde{g}_t^{-1}g_1) \right\|^2_{\mathcal{G}}$$

with $$\tilde{g}_t = g_0 \exp\left(t \log(g_0^{-1} g_1)\right)$$ and $\|\cdot\|_{\mathcal{G}}$ a left-invariant Riemannian metric on $\mathfrak{g}$.

2. Symmetry Discovery and the LieFlow Paradigm

Symmetry discovery is posed as learning the (unknown) symmetry subgroup $H \subseteq G$ that stabilizes a data distribution $q(x)$ under the group action $g \cdot x$. The condition $q(h \cdot x) = q(x)$ for all $h \in H$ defines exact symmetries (Park et al., 23 Dec 2025). The computational task is to learn a distribution $p_\theta(g)$ over $G$ that concentrates on $H$ without explicit prior knowledge of $H$.

LieFlow formulates this as a flow matching process directly on $G$, mapping a tractable prior $p_0$ (e.g., uniform measure on $G$ or a Gaussian on $\mathfrak{g}$, pushed forward via the exponential map) to a target $p_1$ supported on $H$. The learned vector field $v_t^\theta: G \rightarrow \mathfrak{g}$ induces an ODE:

$\dot{g}_t = (L_{g_t})_* v_t^\theta(g_t)$

with the goal that the pushforward of $p_0$ under the solution of this ODE closely approximates $p_1$ as $t \to 1$. The training loss encourages $v_t^\theta$ to match the ground-truth direction to $H$ along exponential curves, measured in the Lie algebra norm.

A core challenge is last-minute mode convergence in the case where $H$ is a finite subgroup: modes corresponding to discrete symmetries (e.g., reflections, rotations) are equally distant for much of the flow, resulting in indecisive dynamics until very late in $t$. LieFlow addresses this by biasing the distribution of $t$ toward late times, e.g., $t \sim \mathrm{Beta}(n,1)$ with $n > 1$, allocating training emphasis near $t \to 1$ to facilitate timely mode discrimination (Park et al., 23 Dec 2025).

3. Algorithmic Formulation and Implementation

The canonical workflow for flow matching on Lie groups proceeds as follows:

• Sampling and Setup: Sample $x_1 \sim q$, $g \sim p_0$, set $x_0 = g \cdot x_1$, compute $A = \log(g^{-1})$.
• Interpolation: Form the exponential-curve interpolant $x_t = \exp(t A) \cdot x_0$.
• Network Architecture: $v_t^\theta$ is typically parameterized as a multi-layer perceptron with $(x_t, t)$ (and possibly $g_1$) as input, outputting an element in $\mathbb{R}^{\dim \mathfrak{g}}$.
• Loss Evaluation: At each minibatch, minimize the mean squared deviation of $v_t^\theta(x_t)$ from $A$ with respect to the left-invariant metric.
• Integration/Sampling: At inference, integrate the ODE with learned $v_t^\theta$ forward in small steps $\Delta t$, accumulating the group element (e.g., via products of exponentials), and mapping back to $G$ as necessary (Sherry et al., 1 Apr 2025, Park et al., 23 Dec 2025, Bertolini et al., 4 Feb 2025).

Implementation requires efficient and stable computation of $\exp: \mathfrak{g} \rightarrow G$ and $\log: G \rightarrow \mathfrak{g}$. For matrix groups (SO(3), GL(2,$\mathbb{C}$), etc.), closed-form solutions (Rodrigues' formula, quaternion representations) or high-order numerical schemes (Padé approximants, Schur-Parlett methods) are used.

4. Experimental Validation and Empirical Findings

Empirical results validate flow matching on Lie groups across both synthetic symmetry discovery tasks and equivariant generative modeling. Representative experiments include:

• 2D Point Cloud Symmetry Discovery: On canonical 2D datasets with known subgroup structure (e.g., $C_4$ or $D_4$ acting on arrows/half-arrows, $G=SO(2)$ or $GL(2,\mathbb{C})$), LieFlow recovered the precise number and location of modes (rotations/reflections) corresponding to group elements. Wasserstein-1 metric evaluated performance; LieFlow achieved $W_1=0.072$ (SO(2)→$C_4$), outperforming previously proposed GAN-based symmetry learners (Park et al., 23 Dec 2025).
• 3D Subgroup Recovery: For point clouds sampled from the orbits of tetrahedral, octahedral, or SO(2) subgroups of $SO(3)$, LieFlow with a power-schedule in $t$ accurately recovered the symmetries (e.g., 24-mode clustering for the octahedral group). For higher order groups (e.g., icosahedral with $|H|=60$), recovery remained challenging.
• Generative Modeling: On $SE(2)$, $SO(3)$, and product groups, flow matching produces visually smooth flows interpolating between target poses or orientations, directly along exponential curves, confirming the geometric soundness of the construction (Sherry et al., 1 Apr 2025). In the generalized score matching setting, learning in Lie algebra coordinates can reduce effective learning dimensionality and enhance sample efficiency (Bertolini et al., 4 Feb 2025).

5. Relationship to Other Geometric and Group-Valued Flows

Flow matching on Lie groups is related to, but distinct from, bi-invariant consensus flows, synchronization dynamics, and Laplacian-based flows on groups, as studied in distributed systems, robotics, and control theory. Consensus flows employ inter-agent "spring energies" and Laplacian couplings, with dynamics of the form:

$$g_i^{-1} \dot{g}_i = -k_P \sum_j w_{ij} \log(g_i^{-1} g_j)$$

to promote alignment over a network on $G$ equipped with a bi-invariant metric (Chandrasekharan et al., 2022). These flows yield convergence to consensus configurations, stability governed by the Laplacian spectrum, and are special cases of the general flow matching principle with collaboration between networked elements rather than direct sampling.

Score-based diffusion and generalized score matching approaches further lift the notion of flows to stochastic settings over Lie groups, with Langevin or SDE trajectories composed in the Lie algebra, yielding generative processes via either deterministic flows (as in flow matching) or stochastic integration (Bertolini et al., 4 Feb 2025). This unification connects flow matching, score-based modeling, and traditional group-valued synchronization.

6. Limitations, Open Problems, and Future Directions

While flow matching on Lie groups is geometrically natural and empirically effective, several challenges remain:

• Mode Discrimination in Highly Symmetric or High-Order Discrete Groups: Mode collapse or last-minute convergence persists with large or complex finite subgroups (e.g., icosahedral group). Skewing $t$ toward late times ameliorates, but does not eliminate, this phenomenon (Park et al., 23 Dec 2025).
• Likelihood and Density Estimation: Current implementations yield "likelihood-free" models—no explicit density is defined for $p_\theta(g)$, so evaluation relies on empirical metrics or histogramming.
• Extension to Manifolds and Homogeneous Spaces: The current framework assumes surjective exponential maps (matrix groups), but extension to coset spaces $G/H$ and more general Riemannian manifolds is an open area (Sherry et al., 1 Apr 2025).
• Robustness to Partial or Noisy Data: Discovery of approximate or latent symmetries in real-world or incomplete datasets motivates further research.
• Optimal ODE Integration and Numerical Stability: Algorithmic accuracy and stability for high-dimensional or stiff Lie group flows, and integration with Riemannian diffusion methods, is under development.

A plausible implication is that integrating flow matching with Riemannian diffusion processes or advanced equilibria techniques may enable better mode separation and sampling efficiency.

7. Synthesis and Impact

Flow matching on Lie groups enables flexible, adaptive modeling of both continuous and discrete symmetries directly in group-theoretic domains. By leveraging intrinsic geometry, it unifies symmetry discovery, generative modeling, and group-valued flow concepts, with demonstrated advantages over prior methods requiring hand-designed augmentations or explicit density modeling. The approach is extensible to a range of matrix groups and offers a principled framework for further research into generative modeling and symmetry-aware machine learning on structured spaces (Park et al., 23 Dec 2025, Sherry et al., 1 Apr 2025, Bertolini et al., 4 Feb 2025).