Symmetry is a core property of physical systems and machine learning models, but identifying symmetries in data automatically remains a difficult problem. A new framework called LieFlow aims to solve this by reframing symmetry discovery as a distribution learning problem on Lie groups, according to a preprint on arXiv.
LieFlow operates directly in group space, modeling a symmetry distribution over a large hypothesis group $G$. The support of the learned distribution reveals the underlying symmetry group $H \subseteq G$. Unlike previous work, LieFlow can discover both continuous and discrete symmetries within a unified framework, without assuming a fixed Lie algebra basis or a specific distribution over group elements.
The Challenge of Symmetry Discovery
Discovering symmetries automatically is challenging because many real-world datasets exhibit a mix of continuous rotations and discrete reflections. Prior methods like LieGAN required prior knowledge of the Lie algebra structure and struggled with discrete symmetries. LieFlow overcomes this by operating in group space rather than searching for generators.
How LieFlow Works
LieFlow employs flow matching on Lie groups to learn a distribution that concentrates on the true symmetry subgroup. This approach does not require a fixed basis, making it flexible for different types of symmetries. The framework is designed to work with high-dimensional group representations, enabling discovery of subgroups from data.
| Feature | LieFlow | LieGAN (baseline) |
|---|---|---|
| Continuous symmetries | Yes | Yes (limited) |
| Discrete symmetries | Yes | Poor |
| Fixed Lie algebra basis | No | Required |
| Unified framework | Yes | No |
According to the paper, LieFlow significantly outperforms LieGAN, a state-of-the-art baseline, in identifying discrete symmetries.
Experimental Validation
The researchers evaluated LieFlow on several datasets: synthetic 2D and 3D point clouds, ModelNet10, and the real-world MI-Motion dataset. In all cases, LieFlow accurately discovered both continuous and discrete subgroups, with particular strength in detecting discrete symmetries where LieGAN failed.
For the target audience of CTOs and technology procurement leaders, symmetry discovery has practical implications. In machine learning, knowing the underlying symmetries can improve sample efficiency and model performance, reducing the amount of labeled data needed for training. This is especially relevant in fields like robotics, autonomous systems, and supply chain logistics, where data labeling is expensive and patterns often exhibit symmetries.
Broader Impact
LieFlow is a research advance, but its unified treatment of continuous and discrete symmetries could extend to applications in physical simulations, drug discovery, and anomaly detection in manufacturing. By automating symmetry discovery, enterprises can integrate more efficient AI models without manual feature engineering.
The preprint, authored by Chen, Yuxuan Park, Jung Yeon Eijkelboom, Floor Yang, Jianke Van De Meent, Jan-Willem Wong, Lawson L S Walters, and Robin, is available on arXiv under a Creative Commons license.