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.

翻译：对称性是理解物理系统的基石，且能提升机器学习中的性能与样本效率。这两项研究都需要掌握数据中潜在对称性的知识，然而自动发现这些对称性却颇具挑战。我们提出LieFlow，一种新颖框架，将对称性发现重构为李群上的分布学习问题。不同于搜索对称生成元，我们的方法直接在群空间中操作，在大型假设群$G$上建模对称性分布。学习到的分布支撑集揭示了潜在对称群$H\subseteq G$。与先前工作不同，LieFlow能在统一框架内同时发现连续对称性与离散对称性，无需假设固定的李代数基或对群元素施加特定分布。在合成2D与3D点云、ModelNet10及真实世界MI-Motion数据集上的实验表明，LieFlow能准确发现连续子群与离散子群，在识别离散对称性方面显著超越当前最先进基线方法LieGAN。