Symmetry-driven graph neural networks

TL;DR

This paper introduces two novel graph neural network architectures that leverage symmetries and invariance properties to improve data efficiency and generalization, specifically focusing on Euclidean and conformal group transformations.

Contribution

The paper presents new equivariant graph neural network architectures that are invariant to Euclidean and conformal transformations, enhancing efficiency and generalization capabilities.

Findings

Models are more data-efficient than classical architectures.

Equivariance improves generalization to geometric transformations.

Limitations arise when the data lacks the assumed symmetries.

Abstract

Exploiting symmetries and invariance in data is a powerful, yet not fully exploited, way to achieve better generalisation with more efficiency. In this paper, we introduce two graph network architectures that are equivariant to several types of transformations affecting the node coordinates. First, we build equivariance to any transformation in the coordinate embeddings that preserves the distance between neighbouring nodes, allowing for equivariance to the Euclidean group. Then, we introduce angle attributes to build equivariance to any angle preserving transformation - thus, to the conformal group. Thanks to their equivariance properties, the proposed models can be vastly more data efficient with respect to classical graph architectures, intrinsically equipped with a better inductive bias and better at generalising. We demonstrate these capabilities on a synthetic dataset composed of $n$-dimensional geometric objects. Additionally, we provide examples of their limitations when (the right) symmetries are not present in the data.