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

Ye Qin, Jingchao Wang, Yang Shi, Haiying Huang, Junxu Li, Weijian Liu, Tinghui Chen, Jinghui Qin

TL;DR

PR-CapsNet introduces a Pseudo-Riemannian Capsule Network that models graph data in adaptive-curvature manifolds to capture mixed geometric structures. It extends capsule routing to geodesically disconnected spaces via a diffeomorphism-based tangent-space routing, and augments routing with multi-perspective, curvature-aware gating to adapt to local geometry. A geometry-preserving classifier operates in tangent space with curvature-weighted decisions, and an efficient routing algorithm enables scalable training. Empirical results on node and graph classification benchmarks demonstrate state-of-the-art performance and robust handling of hierarchical, clustered, and cyclic graph structures. This work advances geometric deep learning by unifying CapsNets with flexible pseudo-Riemannian geometry for complex graphs.

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.

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

TL;DR

PR-CapsNet introduces a Pseudo-Riemannian Capsule Network that models graph data in adaptive-curvature manifolds to capture mixed geometric structures. It extends capsule routing to geodesically disconnected spaces via a diffeomorphism-based tangent-space routing, and augments routing with multi-perspective, curvature-aware gating to adapt to local geometry. A geometry-preserving classifier operates in tangent space with curvature-weighted decisions, and an efficient routing algorithm enables scalable training. Empirical results on node and graph classification benchmarks demonstrate state-of-the-art performance and robust handling of hierarchical, clustered, and cyclic graph structures. This work advances geometric deep learning by unifying CapsNets with flexible pseudo-Riemannian geometry for complex graphs.

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.

Paper Structure

This paper contains 24 sections, 1 theorem, 23 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Pseudo-Riemannian Manifolds
  4. CapsNets in Graph Learning
  5. Preliminary
  6. Pseudo-Riemannian Manifolds
  7. Capsule Networks and Dynamic Routing
  8. PR-CapsNet
  9. Pseudo-Riemannian Capsule Routing
  10. Adaptive Curvature Routing
  11. Geometric Perspectives and Learned Weights.
  12. Curvature-Aware Gating Mechanism.
  13. Dynamic Aggregation of Multi-Perspective Features.
  14. Pseudo-Riemannian Capsule Classifier
  15. Pseudo-Riemannian Routing Algorithm
  16. ...and 9 more sections

Key Result

theorem 1

There exists a diffeomorphism $\psi: \mathcal{Q}_{s,t}^\beta \to S_t^{-\beta} \times \mathbb{R}^s$, where $S_t^{-\beta}$ is a $t$-dimensional sphere of radius $\sqrt{|-\beta|}$ (if $\beta < 0$) or a hyperbolic space (if $\beta > 0$ and $S_t^{-\beta}$ is interpreted accordingly), and $\mathbb{R}^s$ i where $(\mathbf{U}, \mathbf{V}) \in S_t^{-\beta} \times \mathbb{R}^s$. For $\beta > 0$, similar for

Figures (6)

  • Figure 1: t-SNE visualization of PubMed. Dark brown denotes the class of Experimental, light orange denotes Type 1, and light blue denotes Type 2.
  • Figure 2: Ablation study of geometric components on Cora and CiteSeer. "Fixed Curvature" denotes a model with a frozen pseudo-Riemannian metric.
  • Figure 3: Effects of the number of routing iterations.
  • Figure 4: Effects of different spatiotemporal dimensions on test accuracy.
  • Figure 5: Sensitivity analysis of perspectives $K$ on Cora. Performance peaks at $K=4$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • definition 1: Pseudo-Euclidean Space
  • definition 2: Pseudo-hyperboloid $\mathcal{Q}_{s,t}^\beta$
  • theorem 1: Diffeomorphism $\psi$
  • definition 3: Diffeomorphic Logarithmic and Exponential Maps 13