# Unit Ball Model for Embedding Hierarchical Structures in the Complex Hyperbolic Space

arXiv: https://arxiv.org/abs/2105.03966 · HTML: https://mlsanity.com/paper/2105.03966/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 48 references that are in the map.
- First posted on arXiv 9 May 2021.

## Builds on (most cited first, 22 of 22)

- [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)
- [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)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [A Hyperbolic-to-Hyperbolic Graph Convolutional Network](https://mlsanity.com/paper/2104.06942/) (2021)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [Knowledge Association with Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2010.02162/) (2020)
- [Improving Hypernymy Detection with an Integrated Path-based and Distributional Method](https://mlsanity.com/paper/1603.06076/) (2016)
- [APo-VAE: Text Generation in Hyperbolic Space](https://mlsanity.com/paper/2005.00054/) (2020)

## Built on by (most cited first, 1 of 1)

- [Complex Hyperbolic Knowledge Graph Embeddings with Fast Fourier Transform](https://mlsanity.com/paper/2211.03635/) (2022)

## Abstract (arXiv)

Learning the representation of data with hierarchical structures in the hyperbolic space attracts increasing attention in recent years. Due to the constant negative curvature, the hyperbolic space resembles tree metrics and captures the tree-like properties naturally, which enables the hyperbolic embeddings to improve over traditional Euclidean models. However, many real-world hierarchically structured data such as taxonomies and multitree networks have varying local structures and they are not trees, thus they do not ubiquitously match the constant curvature property of the hyperbolic space. To address this limitation of hyperbolic embeddings, we explore the complex hyperbolic space, which has the variable negative curvature, for representation learning. Specifically, we propose to learn the embeddings of hierarchically structured data in the unit ball model of the complex hyperbolic space. The unit ball model based embeddings have a more powerful representation capacity to capture a variety of hierarchical structures. Through experiments on synthetic and real-world data, we show that our approach improves over the hyperbolic embedding models significantly. We also explore the competence of complex hyperbolic geometry on the multitree structure and 1-N structure.