AtlasD: Automatic Local Symmetry Discovery

Existing symmetry discovery methods predominantly focus on global transformations across the entire system or space, but they fail to consider the symmetries in local neighborhoods. This may result in the reported symmetry group being a misrepresentation of the true symmetry. In this paper, we formalize the notion of local symmetry as atlas equivariance. Our proposed pipeline, automatic local symmetry discovery (AtlasD), recovers the local symmetries of a function by training local predictor networks and then learning a Lie group basis to which the predictors are equivariant. We demonstrate AtlasD is capable of discovering local symmetry groups with multiple connected components in top-quark tagging and partial differential equation experiments. The discovered local symmetry is shown to be a useful inductive bias that improves the performance of downstream tasks in climate segmentation and vision tasks. Our code is publicly available at https://github.com/Rose-STL-Lab/AtlasD.

Symmetry discovery is a problem in machine learning where one seeks to find important properties about a dataset in order to build more accurate prediction models. Existing methods for finding symmetries in data look for patterns that apply everywhere uniformly (like how a circle looks the same when rotated about any point). However, many real-world systems have more complex symmetries that vary from place to place, like how ocean currents might have rotational patterns in some regions but not others.We introduce a new framework called AtlasD (Automatic Local Symmetry Discovery) that can detect these "local symmetries". These are simply transformations that preserve structure in specific neighborhoods rather than globally. Our method works by training small neural networks to make predictions in local regions, then automatically discovering what transformations (called Lie groups) these networks are invariant to.We tested AtlasD on challenging problems in particle physics and solving differential equations, successfully finding complex symmetry groups that previous methods missed. When we use these discovered local symmetries as inductive biases in downstream tasks, we see significant performance improvements. This work opens up possibilities for understanding the hidden structure in complex datasets and leveraging these insights to build more effective machine learning models.