Discovering Symmetry Groups with Flow Matching

TL;DR

LieFlow is a novel framework that automatically discovers both continuous and discrete symmetries in data by modeling a distribution over Lie groups, improving symmetry detection in physical and machine learning systems.

Contribution

It introduces a unified approach to symmetry discovery that operates directly in group space without fixed bases, outperforming previous methods like LieGAN.

Findings

Accurately discovers continuous and discrete symmetries in synthetic data.

Significantly outperforms LieGAN in identifying discrete symmetries.

Works effectively on 2D, 3D point clouds, and ModelNet10 datasets.

Abstract

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 and ModelNet10 show that LieFlow accurately discovers continuous and discrete subgroups, significantly outperforming a state-of-the-art baseline, LieGAN, in identifying discrete symmetries.