AtlasD: Automatic Local Symmetry Discovery

2025-04-15

AI Summary

Existing symmetry discovery methods model only global transformations, neglecting local neighborhood structure and thus prone to misidentifying symmetry groups. This work formalizes local symmetry as atlas-equivariance—the first such formulation—and introduces the first automated framework for discovering local symmetries. Our method identifies local symmetry groups across multiple connected components; integrates local symmetry predictors, Lie group basis learning, and equivariance-constrained optimization; and embeds local symmetry as an inductive bias into function-space modeling. We successfully uncover intricate local symmetry structures in top-quark tagging and PDE solving. Moreover, incorporating local symmetry as an inductive bias significantly improves downstream model accuracy on climate segmentation and vision benchmarks. These results empirically validate local symmetry as a powerful inductive bias for geometric deep learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Local Search
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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.

Problem

• Identifies local symmetries overlooked by global methods
• Proposes AtlasD to discover local symmetries via predictor networks
• Demonstrates local symmetry benefits in climate and vision tasks

Innovation

• Formalizes local symmetry as atlas equivariance
• Trains local predictor networks for symmetry
• Learns Lie group basis for equivariant predictors