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

arXiv: https://arxiv.org/abs/2512.08218 · HTML: https://mlsanity.com/paper/2512.08218/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 37 of its 68 references that are in the map.
- First posted on arXiv 9 Dec 2025.

## Builds on (most cited first, 30 of 37)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Learning Convolutional Neural Networks for Graphs](https://mlsanity.com/paper/1605.05273/) (2016)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Classic GNNs are Strong Baselines: Reassessing GNNs for Node Classification](https://mlsanity.com/paper/2406.08993/) (2024)
- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [Neural Embeddings of Graphs in Hyperbolic Space](https://mlsanity.com/paper/1705.10359/) (2017)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [Spherical Text Embedding](https://mlsanity.com/paper/1911.01196/) (2019)
- [Dynamic Context-guided Capsule Network for Multimodal Machine Translation](https://mlsanity.com/paper/2009.02016/) (2020)
- [COVID-CAPS: A Capsule Network-based Framework for Identification of COVID-19 cases from X-ray Images](https://mlsanity.com/paper/2004.02696/) (2020)
- [Self-Supervised Continual Graph Learning in Adaptive Riemannian Spaces](https://mlsanity.com/paper/2211.17068/) (2022)
- [Network Geometry](https://mlsanity.com/paper/2001.03241/) (2020)
- [Pseudo-Riemannian Graph Convolutional Networks](https://mlsanity.com/paper/2106.03134/) (2021)
- [Investigating Capsule Networks with Dynamic Routing for Text Classification](https://mlsanity.com/paper/1804.00538/) (2018)
- [Ultrahyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2206.00449/) (2022)
- [Ultrahyperbolic Representation Learning](https://mlsanity.com/paper/2007.00211/) (2020)
- [A Capsule Network for Recommendation and Explaining What You Like and Dislike](https://mlsanity.com/paper/1907.00687/) (2019)
- [DeepSphere: towards an equivariant graph-based spherical CNN](https://mlsanity.com/paper/1904.05146/) (2019)

## Abstract (arXiv)

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.