# Low-distortion and GPU-compatible Tree Embeddings in Hyperbolic Space

arXiv: https://arxiv.org/abs/2502.17130 · HTML: https://mlsanity.com/paper/2502.17130/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 60 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Builds on (most cited first, 25 of 25)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [Embedding Text in Hyperbolic Spaces](https://mlsanity.com/paper/1806.04313/) (2018)
- [Learning towards Minimum Hyperspherical Energy](https://mlsanity.com/paper/1805.09298/) (2018)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Inferring Concept Hierarchies from Text Corpora via Hyperbolic Embeddings](https://mlsanity.com/paper/1902.00913/) (2019)
- [Hierarchical Image Classification using Entailment Cone Embeddings](https://mlsanity.com/paper/2004.03459/) (2020)
- [Hyperbolic Busemann Learning with Ideal Prototypes](https://mlsanity.com/paper/2106.14472/) (2021)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [HyperKG: Hyperbolic Knowledge Graph Embeddings for Knowledge Base Completion](https://mlsanity.com/paper/1908.04895/) (2019)
- [Skin Lesion Recognition with Class-Hierarchy Regularized Hyperbolic Embeddings](https://mlsanity.com/paper/2209.05842/) (2022)
- [From Isolated Islands to Pangea: Unifying Semantic Space for Human Action Understanding](https://mlsanity.com/paper/2304.00553/) (2023)
- [Hyperbolic Audio-visual Zero-shot Learning](https://mlsanity.com/paper/2308.12558/) (2023)
- [Exploring Data Geometry for Continual Learning](https://mlsanity.com/paper/2304.03931/) (2023)
- [Hyperbolic Audio Source Separation](https://mlsanity.com/paper/2212.05008/) (2022)
- [Hierarchical Explanations for Video Action Recognition](https://mlsanity.com/paper/2301.00436/) (2023)
- [MCTensor: A High-Precision Deep Learning Library with Multi-Component Floating-Point](https://mlsanity.com/paper/2207.08867/) (2022)

## Built on by (most cited first, 6 of 6)

- [PHyCLIP: ℓ1-Product of Hyperbolic Factors Unifies Hierarchy and Compositionality in Vision-Language Representation Learning](https://mlsanity.com/paper/2510.08919/) (2025)
- [Minimizing Hyperbolic Embedding Distortion with LLM-Guided Hierarchy Restructuring](https://mlsanity.com/paper/2511.20679/) (2025)
- [Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity](https://mlsanity.com/paper/2505.21073/) (2025)
- [HyperHELM: Hyperbolic Hierarchy Encoding for mRNA Language Modeling](https://mlsanity.com/paper/2509.24655/) (2025)
- [Real-Time Hybrid Retrieval in Hyperbolic Space for Retrieval-Augmented Generation on Edge Devices](https://mlsanity.com/paper/2608.01450/) (2026)
- [HyperANFIS: Enhancing Rule Representation and Interpretability in Adaptive Neuro-Fuzzy Systems via Hyperbolic Geometry](https://mlsanity.com/paper/2608.11768/) (2026)

## Abstract (arXiv)

Embedding tree-like data, from hierarchies to ontologies and taxonomies, forms a well-studied problem for representing knowledge across many domains. Hyperbolic geometry provides a natural solution for embedding trees, with vastly superior performance over Euclidean embeddings. Recent literature has shown that hyperbolic tree embeddings can even be placed on top of neural networks for hierarchical knowledge integration in deep learning settings. For all applications, a faithful embedding of trees is needed, with combinatorial constructions emerging as the most effective direction. This paper identifies and solves two key limitations of existing works. First, the combinatorial construction hinges on finding highly separated points on a hypersphere, a notoriously difficult problem. Current approaches achieve poor separation, degrading the quality of the corresponding hyperbolic embedding. We propose highly separated Delaunay tree embeddings (HS-DTE), which integrates angular separation in a generalized formulation of Delaunay embeddings, leading to lower embedding distortion. Second, low-distortion requires additional precision. The current approach for increasing precision is to use multiple precision arithmetic, which renders the embeddings useless on GPUs in deep learning settings. We reformulate the combinatorial construction using floating point expansion arithmetic, leading to superior embedding quality while retaining utility on accelerated hardware.