AdS-GNN -- a Conformally Equivariant Graph Neural Network

TL;DR

This paper introduces AdS-GNN, a graph neural network that is conformally equivariant by lifting data to AdS space, enabling it to leverage conformal symmetries for improved performance in vision and physics tasks.

Contribution

We develop a novel conformally equivariant GNN by lifting data to AdS space, exploiting isometric transformations, and conditioning message-passing on proper distance for efficiency.

Findings

Strong performance on vision and physics tasks

Enhanced generalization capabilities

Ability to extract conformal data such as scaling dimensions

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.