Discovering Lie Groups with Flow Matching
This work addresses the need for automated symmetry discovery in physical systems and machine learning, offering a more flexible approach than previous methods, though it appears incremental in its improvements.
The paper tackles the problem of discovering symmetries in data by proposing a method to learn Lie groups directly from data using flow matching, which successfully identifies discrete groups like reflections in 2D and 3D point clouds. It addresses a challenge called 'last-minute convergence' with a novel interpolation scheme.
Symmetry is fundamental to understanding physical systems, and at the same time, can improve performance and sample efficiency in machine learning. Both pursuits require knowledge of the underlying symmetries in data. To address this, we propose learning symmetries directly from data via flow matching on Lie groups. We formulate symmetry discovery as learning a distribution over a larger hypothesis group, such that the learned distribution matches the symmetries observed in data. Relative to previous works, our method, \lieflow, is more flexible in terms of the types of groups it can discover and requires fewer assumptions. Experiments on 2D and 3D point clouds demonstrate the successful discovery of discrete groups, including reflections by flow matching over the complex domain. We identify a key challenge where the symmetric arrangement of the target modes causes ``last-minute convergence,'' where samples remain stationary until relatively late in the flow, and introduce a novel interpolation scheme for flow matching for symmetry discovery.