PR-CapsNet: Pseudo-Riemannian Capsule Network with Adaptive Curvature Routing for Graph Learning

Capsule Networks (CapsNets) show exceptional graph representation capacity via dynamic routing and vectorized hierarchical representations, but they model the complex geometries of real-world graphs (e.g., hierarchies, clusters, cycles) poorly by fixed-curvature space due to the inherent geodesical disconnectedness issues, leading to suboptimal performance. Recent works find that non-Euclidean pseudo-Riemannian manifolds provide specific inductive biases for embedding graph data, but how to leverage them to improve CapsNets is still underexplored. Here, we extend the Euclidean capsule routing into geodesically disconnected pseudo-Riemannian manifolds and derive a Pseudo-Riemannian Capsule Network (PR-CapsNet), which models data in pseudo-Riemannian manifolds of adaptive curvature, for graph representation learning. Specifically, PR-CapsNet enhances the CapsNet with Adaptive Pseudo-Riemannian Tangent Space Routing by utilizing pseudo-Riemannian geometry. Unlike single-curvature or subspace-partitioning methods, PR-CapsNet concurrently models hierarchical and cluster/cyclic graph structures via its versatile pseudo-Riemannian metric. It first deploys Pseudo-Riemannian Tangent Space Routing to decompose capsule states into spherical-temporal and Euclidean-spatial subspaces with diffeomorphic transformations. Then, an Adaptive Curvature Routing is developed to adaptively fuse features from different curvature spaces for complex graphs via a learnable curvature tensor with geometric attention from local manifold properties. Finally, a geometric properties-preserved Pseudo-Riemannian Capsule Classifier is developed to project capsule embeddings to tangent spaces and use curvature-weighted softmax for classification. Extensive experiments on node and graph classification benchmarks show PR-CapsNet outperforms state-of-the-art models, validating PR-CapsNet's strong representation power for complex graph structures.