AdS-GNN -- a Conformally Equivariant Graph Neural Network

TL;DR

AdS-GNN introduces a conformally equivariant graph neural network by lifting boundary data from $\mathbb{R}^d$ into AdS$_{d+1}$ and performing message passing with AdS proper distance. The approach enforces invariance under the global conformal group via bulk–boundary geometry, while embedding details preserve scale invariance and control symmetry breaking. Empirical results on SuperPixel MNIST, shape segmentation, and 2D Ising correlators show competitive performance, strong scale generalization, and the ability to learn conformal dimensions $\Delta$ that align with theoretical values. The work offers a principled route to exploit conformal symmetry in geometric learning, with implications for critical phenomena and robust shape analysis, though it acknowledges limitations in uplift-induced breaking of special conformal transformations and a focus on scalar features.

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.

Figures (9)

• Figure 1: AdS-GNN lifts points from Euclidean space to Anti de Sitter space and computes message passing conditioned on the proper distance.
• Figure 2: Example of hyperboloid $\|Y\| = -1$ embedded in $\mathbb{R}^{2,1}$, constituting two copies of $\mathop{\mathrm{AdS}}\nolimits_2$.
• Figure 3: An example of the embedding of a boundary digit into $\mathop{\mathrm{AdS}}\nolimits$.
• Figure 4: Test error on augmented data, SuperPixel MNIST.
• Figure 5: Performance on shape segmentation and the Ising task. Left, test accuracy of shape segmentation as a function of the number of training points. Middle, relative L2 as a function of the number of training points, with system size fixed at $N = 16$. Right, loss as a function of system size with 8192 training points. Inset shows $N = 2$.

Theorems & Definitions (46)

• Lemma D.1.1
• proof
• Definition D.1.2: Conformal maps/conformal transformations
• Definition D.1.3: Conformal diffeomorphisms and isometries
• Example D.1.5: Affine conformal diffeomorphisms
• proof
• Theorem D.1.6: Affine conformal diffeomorphisms, see Ami67
• proof
• Definition D.1.7: The linear and affine conformal group
• Example D.1.8: The inversion at the pseudo-sphere