用流匹配发现数据中的连续与离散对称性，提升机器学习效率。
Discovering Symmetry Groups with Flow Matching
- 将对称性发现转化为李群上的分布学习问题，直接在群空间建模。
- 在合成点云和真实运动数据上准确识别连续与离散对称子群。
- 统一框架无需预设基或分布假设，优于现有最佳方法LieGAN。
对称性是理解物理系统的基础，也能提升机器学习的性能与采样效率。二者都依赖于对数据中潜在对称性的认知，但自动发现这些对称性仍具挑战。本文提出LieFlow，一种将对称性发现重构为李群上分布学习的新框架。不直接搜索对称生成元，而是直接在大假设群 $G$ 的群空间中建模对称性分布，其支撑集揭示出底层对称群 $H \⊆ G$。与以往工作不同，LieFlow可在统一框架内发现连续与离散对称性，无需预设李代数基或群元素分布。在二维、三维合成点云，ModelNet10及真实世界MI-Motion数据集上的实验表明，该方法能准确发现连续与离散子群，在离散对称性识别上显著优于当前最优基线LieGAN。
原文摘要 · Abstract (English)
Symmetry is fundamental to understanding physical systems and can improve performance and sample efficiency in machine learning. Both pursuits require knowledge of the underlying symmetries in data, yet discovering these symmetries automatically is challenging. We propose LieFlow, a novel framework that reframes symmetry discovery as a distribution learning problem on Lie groups. Instead of searching for the symmetry generators, our approach 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 works, LieFlow can discover both continuous and discrete symmetries within a unified framework, without assuming a fixed Lie algebra basis or a specific distribution over the group elements. Experiments on synthetic 2D and 3D point clouds, ModelNet10 and a real-world MI-Motion dataset show that LieFlow accurately discovers continuous and discrete subgroups, significantly outperforming a state-of-the-art baseline, LieGAN, in identifying discrete symmetries.
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