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

2025-12-08

AI Summary

Capsule Networks (CapsNets) face fundamental limitations in graph representation learning due to the geodesic incompleteness of fixed-curvature spaces, hindering faithful modeling of complex geometric structures in real-world graphs. To address this, we propose the first CapsNet extension to adaptive-curvature pseudo-Riemannian manifolds. Our method introduces a pseudo-Riemannian tangent-space dynamic routing mechanism that jointly incorporates diffeomorphic transformations, learnable curvature tensors, and geometric attention. It further employs a curvature-weighted softmax classifier to preserve intrinsic geometric properties while enabling unified hierarchical, clustering, and cyclic structural modeling. Extensive experiments demonstrate that our approach achieves significant improvements over state-of-the-art models on both node and graph classification benchmarks. These results empirically validate that adaptive non-Euclidean geometric priors fundamentally enhance graph representation capacity.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

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 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 or 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 SOTA models, validating PR-CapsNet's strong representation power for complex graph structures.

Problem

• Improving graph representation learning with adaptive pseudo-Riemannian manifolds
• Addressing geodesic disconnectedness in capsule networks for graphs
• Enhancing hierarchical and cyclic graph structure modeling via curvature adaptation

Innovation

• Extends capsule routing into pseudo-Riemannian manifolds with adaptive curvature
• Deploys tangent space routing to decompose capsule states into subspaces
• Uses adaptive curvature routing to fuse features from different curvature spaces