AdS-GNN -- a Conformally Equivariant Graph Neural Network

2025-05-19

AI Summary

This work addresses the challenge of constructing graph neural networks (GNNs) that are strictly equivariant under the full conformal group. We propose the first fully conformally equivariant GNN: it lifts Euclidean inputs into anti-de Sitter (AdS) space, where conformal transformations become isometries; leveraging this geometric correspondence, we design a message-passing mechanism based on the intrinsic AdS distance, ensuring exact geometric equivariance. Key contributions include: (1) the first GNN architecture provably equivariant under the complete conformal group; (2) a rigorous conformal–isometric geometric correspondence formalized and integrated into GNN design; and (3) direct analytical extraction of physically interpretable scaling dimensions. Experiments demonstrate significant generalization improvements on computer vision and statistical physics benchmarks, and successful recovery of conformal invariants validates both theoretical soundness and practical efficacy.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science
• Security and Privacy: Large-scale security measurements

Abstract

Conformal symmetries, i.e. coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Problem

• Builds conformally equivariant graph neural network
• Lifts data to Anti de Sitter space for conformal symmetry
• Validates model on computer vision and physics tasks

Innovation

• Lifts data to Anti de Sitter space
• Uses conformally equivariant message-passing layers
• Exploits isometric transformations correspondence