# Ultrahyperbolic Knowledge Graph Embeddings

arXiv: https://arxiv.org/abs/2206.00449 · HTML: https://mlsanity.com/paper/2206.00449/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 39 references that are in the map.
- First posted on arXiv 1 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [Embedding Entities and Relations for Learning and Inference in Knowledge Bases](https://mlsanity.com/paper/1412.6575/) (2014)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [SimplE Embedding for Link Prediction in Knowledge Graphs](https://mlsanity.com/paper/1802.04868/) (2018)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Canonical Tensor Decomposition for Knowledge Base Completion](https://mlsanity.com/paper/1806.07297/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Unifying Knowledge Graph Learning and Recommendation: Towards a Better Understanding of User Preferences](https://mlsanity.com/paper/1902.06236/) (2019)
- [Analogical Inference for Multi-Relational Embeddings](https://mlsanity.com/paper/1705.02426/) (2017)
- [Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Cones](https://mlsanity.com/paper/2110.14923/) (2021)
- [5* Knowledge Graph Embeddings with Projective Transformations](https://mlsanity.com/paper/2006.04986/) (2020)
- [Ultrahyperbolic Representation Learning](https://mlsanity.com/paper/2007.00211/) (2020)
- [Hyperbolic Geometry is Not Necessary: Lightweight Euclidean-Based Models for Low-Dimensional Knowledge Graph Embeddings](https://mlsanity.com/paper/2103.14930/) (2021)
- [Directed Graph Embeddings in Pseudo-Riemannian Manifolds](https://mlsanity.com/paper/2106.08678/) (2021)
- [Embedding Graphs in Lorentzian Spacetime](https://mlsanity.com/paper/1602.03103/) (2016)
- [On Procrustes Analysis in Hyperbolic Space](https://mlsanity.com/paper/2102.03723/) (2021)
- [Irreflexive and Hierarchical Relations as Translations](https://mlsanity.com/paper/1304.7158/) (2013)

## Built on by (most cited first, 20 of 20)

- [Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces](https://mlsanity.com/paper/2211.03536/) (2022)
- [Self-Supervised Continual Graph Learning in Adaptive Riemannian Spaces](https://mlsanity.com/paper/2211.17068/) (2022)
- [Hyperbolic Representation Learning: Revisiting and Advancing](https://mlsanity.com/paper/2306.09118/) (2023)
- [UUKG: Unified Urban Knowledge Graph Dataset for Urban Spatiotemporal Prediction](https://mlsanity.com/paper/2306.11443/) (2023)
- [Normalizing Flow-based Neural Process for Few-Shot Knowledge Graph Completion](https://mlsanity.com/paper/2304.08183/) (2023)
- [Link Prediction with Attention Applied on Multiple Knowledge Graph Embedding Models](https://mlsanity.com/paper/2302.06229/) (2023)
- [Contrastive Graph Clustering in Curvature Spaces](https://mlsanity.com/paper/2305.03555/) (2023)
- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [A Survey on Temporal Knowledge Graph Completion: Taxonomy, Progress, and Prospects](https://mlsanity.com/paper/2308.02457/) (2023)
- [Shrinking Embeddings for Hyper-Relational Knowledge Graphs](https://mlsanity.com/paper/2306.02199/) (2023)
- [Knowledge Graph Embedding: An Overview](https://mlsanity.com/paper/2309.12501/) (2023)
- [Geometric Relational Embeddings: A Survey](https://mlsanity.com/paper/2304.11949/) (2023)
- [Generating SROI^- Ontologies via Knowledge Graph Query Embedding Learning](https://mlsanity.com/paper/2407.09212/) (2024)
- [Contrastive Sequential Interaction Network Learning on Co-Evolving Riemannian Spaces](https://mlsanity.com/paper/2401.01243/) (2024)
- [Fully Hyperbolic Rotation for Knowledge Graph Embedding](https://mlsanity.com/paper/2411.03622/) (2024)
- [Knowledge Graph Embeddings: A Comprehensive Survey on Capturing Relation Properties](https://mlsanity.com/paper/2410.14733/) (2024)
- [Two-dimensional Taxonomy for N-ary Knowledge Representation Learning Methods](https://mlsanity.com/paper/2506.05626/) (2025)
- [HyperComplEx: Adaptive Multi-Space Knowledge Graph Embeddings](https://mlsanity.com/paper/2511.10842/) (2025)
- [PR-CapsNet: Pseudo-Riemannian Capsule Network with Adaptive Curvature Routing for Graph Learning](https://mlsanity.com/paper/2512.08218/) (2025)
- [Relations Are Channels: Knowledge Graph Embedding via Kraus Decompositions](https://mlsanity.com/paper/2605.10317/) (2026)

## Abstract (arXiv)

Recent knowledge graph (KG) embeddings have been advanced by hyperbolic geometry due to its superior capability for representing hierarchies. The topological structures of real-world KGs, however, are rather heterogeneous, i.e., a KG is composed of multiple distinct hierarchies and non-hierarchical graph structures. Therefore, a homogeneous (either Euclidean or hyperbolic) geometry is not sufficient for fairly representing such heterogeneous structures. To capture the topological heterogeneity of KGs, we present an ultrahyperbolic KG embedding (UltraE) in an ultrahyperbolic (or pseudo-Riemannian) manifold that seamlessly interleaves hyperbolic and spherical manifolds. In particular, we model each relation as a pseudo-orthogonal transformation that preserves the pseudo-Riemannian bilinear form. The pseudo-orthogonal transformation is decomposed into various operators (i.e., circular rotations, reflections and hyperbolic rotations), allowing for simultaneously modeling heterogeneous structures as well as complex relational patterns. Experimental results on three standard KGs show that UltraE outperforms previous Euclidean- and hyperbolic-based approaches.