# Complex Hyperbolic Knowledge Graph Embeddings with Fast Fourier Transform

arXiv: https://arxiv.org/abs/2211.03635 · HTML: https://mlsanity.com/paper/2211.03635/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 7 Nov 2022.

## Builds on (most cited first, 19 of 19)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [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 Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [A Review of Relational Machine Learning for Knowledge Graphs](https://mlsanity.com/paper/1503.00759/) (2015)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [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)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [On the Equivalence of Holographic and Complex Embeddings for Link Prediction](https://mlsanity.com/paper/1702.05563/) (2017)
- [Kernel Cross-Correlator](https://mlsanity.com/paper/1709.05936/) (2017)
- [Hyperbolic Geometry is Not Necessary: Lightweight Euclidean-Based Models for Low-Dimensional Knowledge Graph Embeddings](https://mlsanity.com/paper/2103.14930/) (2021)
- [A Non-commutative Bilinear Model for Answering Path Queries in Knowledge Graphs](https://mlsanity.com/paper/1909.01567/) (2019)
- [Unit Ball Model for Embedding Hierarchical Structures in the Complex Hyperbolic Space](https://mlsanity.com/paper/2105.03966/) (2021)

## Built on by (most cited first, 5 of 5)

- [Hyperbolic Large Language Models](https://mlsanity.com/paper/2509.05757/) (2025)
- [Accelerating hyperbolic t-SNE](https://mlsanity.com/paper/2401.13708/) (2024)
- [Fully Hyperbolic Rotation for Knowledge Graph Embedding](https://mlsanity.com/paper/2411.03622/) (2024)
- [HyperComplEx: Adaptive Multi-Space Knowledge Graph Embeddings](https://mlsanity.com/paper/2511.10842/) (2025)
- [Batch Normalization for Neural Networks on Complex Domains](https://mlsanity.com/paper/2605.00467/) (2026)

## Abstract (arXiv)

The choice of geometric space for knowledge graph (KG) embeddings can have significant effects on the performance of KG completion tasks. The hyperbolic geometry has been shown to capture the hierarchical patterns due to its tree-like metrics, which addressed the limitations of the Euclidean embedding models. Recent explorations of the complex hyperbolic geometry further improved the hyperbolic embeddings for capturing a variety of hierarchical structures. However, the performance of the hyperbolic KG embedding models for non-transitive relations is still unpromising, while the complex hyperbolic embeddings do not deal with multi-relations. This paper aims to utilize the representation capacity of the complex hyperbolic geometry in multi-relational KG embeddings. To apply the geometric transformations which account for different relations and the attention mechanism in the complex hyperbolic space, we propose to use the fast Fourier transform (FFT) as the conversion between the real and complex hyperbolic space. Constructing the attention-based transformations in the complex space is very challenging, while the proposed Fourier transform-based complex hyperbolic approaches provide a simple and effective solution. Experimental results show that our methods outperform the baselines, including the Euclidean and the real hyperbolic embedding models.