Discovering Symmetry Groups with Flow Matching

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.

Many scientific systems have hidden patterns. For example, rotating a molecule or a 3D object may not change its essential properties. These patterns, called symmetries, help scientists understand physical systems and help machine learning models learn more efficiently. However, in many real-world problems, we do not know the correct symmetry in advance, and manually identifying it can be difficult. We introduce LieFlow, a method that automatically discovers symmetries from data. Instead of asking a model to directly guess a small set of symmetry rules, LieFlow searches over a larger space of possible transformations and learns which transformations keep the data looking valid. It does this using a modern generative modeling technique called flow matching, adapted to work directly over transformation groups. This allows the method to find both smooth continuous symmetries, such as rotations by any angle, and discrete symmetries, such as rotations by only 90 degree steps. Our experiments show that LieFlow can recover hidden symmetries in synthetic shapes, 3D objects, and real-world motion data. This could help researchers build machine learning systems that better understand structure in scientific and geometric data.