Cutting-Edge AI Research for Enterprise

The Foundation: Adaptive Pseudo-Riemannian Manifolds

Pseudo-Riemannian geometry extends traditional Riemannian geometry by allowing indefinite metric tensors, enabling manifolds to possess varying local geometric properties. Unlike fixed-curvature spaces, these manifolds can concurrently model both positive (spherical/cluster-like) and negative (hyperbolic/hierarchical) curvatures within a single, unified space.

A critical innovation is the use of diffeomorphic transformations, which map geodesically disconnected pseudo-hyperboloids to geodesically connected product spaces. This mathematical elegance enables well-defined geometric operations, such as logarithmic and exponential maps, which are crucial for consistent and robust feature aggregation in our PR-CapsNet.

PR-CapsNet: Dynamic Routing with Adaptive Curvature

The core of PR-CapsNet lies in its novel Adaptive Curvature Routing (ACR) mechanism. It extends Euclidean capsule routing into pseudo-Riemannian manifolds, allowing capsules to dynamically adapt to the local geometry of the graph data. Key steps include:

• Pseudo-Riemannian Tangent Space Routing: Decomposes capsule states into spherical-temporal and Euclidean-spatial subspaces, resolving geodesic disconnectedness.
• Adaptive Curvature Routing: Employs a learnable curvature tensor and geometric attention to adaptively fuse features from different curvature spaces, ensuring optimal alignment with complex graph structures.
• Pseudo-Riemannian Capsule Classifier: Projects capsule embeddings to tangent spaces and uses curvature-weighted softmax for classification, preserving geometric properties.

This architecture allows PR-CapsNet to concurrently model diverse graph structures with unmatched flexibility and accuracy.

Validating Superiority Across Benchmarks

Extensive experiments on widely recognized node and graph classification benchmarks, including Cora, Citeseer, PubMed, MUTAG, and PROTEINS, consistently demonstrate PR-CapsNet's superior performance over state-of-the-art models.

For instance, PR-CapsNet achieved an absolute accuracy improvement of 6.4 percentage points over the best-performing baseline on the Citeseer node classification dataset. Qualitative analyses, such as t-SNE visualizations, further confirm that PR-CapsNet produces significantly more class-discriminative and compact embeddings, highlighting its robust representation power for intricate graph structures.