Discovering Lie Groups with Flow Matching

2025-12-22

AI Summary

This work addresses the challenge of missing symmetry priors in physical system modeling and machine learning by proposing a novel paradigm for directly discovering latent symmetry structures from high-dimensional data (e.g., 2D/3D point clouds). Methodologically, it formulates symmetry discovery as a flow matching problem on Lie groups and introduces LieFlow—a unified framework capable of handling both continuous and discrete symmetry groups, including non-compact and disconnected groups such as complex-domain reflections. To overcome convergence failure of conventional flow matching at critical time points, LieFlow introduces a novel complex-domain interpolation strategy. Experiments demonstrate that LieFlow accurately identifies rotational and reflective symmetries, significantly improving distribution matching accuracy and robustness in symmetry identification—even under few-shot settings. The framework provides a learnable, generalizable tool for group-structure discovery, enabling symmetry-driven generative modeling and physics-informed learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Symmetry is fundamental to understanding physical systems, and at the same time, can improve performance and sample efficiency in machine learning. Both pursuits require knowledge of the underlying symmetries in data. To address this, we propose learning symmetries directly from data via flow matching on Lie groups. We formulate symmetry discovery as learning a distribution over a larger hypothesis group, such that the learned distribution matches the symmetries observed in data. Relative to previous works, our method, lieflow, is more flexible in terms of the types of groups it can discover and requires fewer assumptions. Experiments on 2D and 3D point clouds demonstrate the successful discovery of discrete groups, including reflections by flow matching over the complex domain. We identify a key challenge where the symmetric arrangement of the target modes causes ``last-minute convergence,'' where samples remain stationary until relatively late in the flow, and introduce a novel interpolation scheme for flow matching for symmetry discovery.

Problem

• Learn symmetries directly from data via flow matching on Lie groups.
• Discover discrete groups like reflections using flow matching over complex domains.
• Address 'last-minute convergence' challenge with a novel interpolation scheme.

Innovation

• Flow matching on Lie groups for symmetry discovery
• Learning distribution over hypothesis group from data
• Novel interpolation scheme addressing last-minute convergence