AdS-GNN -- a Conformally Equivariant Graph Neural Network
This addresses the need for conformally equivariant models in fields like physics and computer vision, representing a novel method for a known bottleneck rather than an incremental improvement.
The authors tackled the problem of building a neural network equivariant under general conformal transformations by lifting data from Euclidean to Anti de Sitter space, resulting in a model that demonstrated strong performance, improved generalization, and the ability to extract conformal data like scaling dimensions in tasks from computer vision and statistical physics.
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.