[Paper Review] Constant Curvature Graph Convolutional Networks

Gregor Bachmann, Gary Bécigneul+1 more|arXiv (Cornell University)|Nov 12, 2019

Topological and Geometric Data Analysis56 references35 citations

TL;DR

The paper generalizes graph convolutional networks to operate in spaces of constant curvature (positive, negative, and zero) using a unified kappa-stereographic gyroframework, enabling differentiable interpolation across curvatures and improved node representation for non-Euclidean graph data.

ABSTRACT

Interest has been rising lately towards methods representing data in non-Euclidean spaces, e.g. hyperbolic or spherical, that provide specific inductive biases useful for certain real-world data properties, e.g. scale-free, hierarchical or cyclical. However, the popular graph neural networks are currently limited in modeling data only via Euclidean geometry and associated vector space operations. Here, we bridge this gap by proposing mathematically grounded generalizations of graph convolutional networks (GCN) to (products of) constant curvature spaces. We do this by i) introducing a unified formalism that can interpolate smoothly between all geometries of constant curvature, ii) leveraging gyro-barycentric coordinates that generalize the classic Euclidean concept of the center of mass. Our class of models smoothly recover their Euclidean counterparts when the curvature goes to zero from either side. Empirically, we outperform Euclidean GCNs in the tasks of node classification and distortion minimization for symbolic data exhibiting non-Euclidean behavior, according to their discrete curvature.

Motivation & Objective

• Motivate why non-Euclidean geometries (hyperbolic, spherical) can better model certain graph-structured data than Euclidean space.
• Introduce a unified gyrovector-space framework that interpolates between constant-curvature geometries.
• Develop kappa-GCNs that operate in the kappa-stereographic model and recover Euclidean GCNs as curvature approaches zero.
• Enable differentiable learning of curvature and geometry jointly with graph neural network training.

Proposed method

• Define the kappa-stereographic model to unify positive and negative curvature in a single framework.
• Extend gyrovector space operations (kappa-addition, kappa-scaling) with closed-form expressions for distance, exp/log maps, and geodesics.
• Introduce kappa-right-matrix-multiplication as a curvature-aware linear transform of embeddings.
• Propose kappa-left-matrix-multiplication as a weighted gyromidpoint-based aggregation for message passing.
• Adopt kappa-logit layer and curvature-aware softmax for classification tasks.
• Prove key properties: intrinsicity of left-multiplication with right-stochastic matrices, and differentiable curvature interpolation.

Experimental results

Research questions

• RQ1 Can constant-curvature spaces provide a differentiable interpolation between Euclidean, hyperbolic, and spherical geometries for GCNs?
• RQ2 Do kappa-GCNs improve node classification and distortion minimization on non-Euclidean graphs relative to Euclidean GCNs and prior hyperbolic methods?
• RQ3 Does learning curvature improve representations for graphs with non-Euclidean structure?
• RQ4 Do product spaces of constant-curvature components (e.g., H^8 × S^8) yield further gains over single-geometry embeddings?

Key findings

• κ-GCNs outperform Euclidean GCNs on node classification datasets and distortion minimization on synthetic and real graphs.
• The model recovers Euclidean GCNs in the limit κ → 0, ensuring a continuous interpolation between geometries.
• Product-geometry embeddings (e.g., hyperbolic or spherical components) can further reduce distortion and improve performance on certain datasets.
• Datasets studied (Citeseer, Cora, Pubmed, Airport) show κ-GCNs achieving competitive or superior accuracy compared with Euclidean and prior hyperbolic GNNs.
• The κ-stereographic framework provides closed-form operations for distance, exp/log maps, and geodesics, enabling efficient differentiable learning across curvatures.