# Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding

arXiv: https://arxiv.org/abs/2005.03847 · HTML: https://mlsanity.com/paper/2005.03847/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2020.

## Built on by (most cited first, 30 of 36)

- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [A Hyperbolic-to-Hyperbolic Graph Convolutional Network](https://mlsanity.com/paper/2104.06942/) (2021)
- [Hyperbolic Representation Learning: Revisiting and Advancing](https://mlsanity.com/paper/2306.09118/) (2023)
- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [No Metric to Rule Them All: Toward Principled Evaluations of Graph-Learning Datasets](https://mlsanity.com/paper/2502.02379/) (2025)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)
- [Random Laplacian Features for Learning with Hyperbolic Space](https://mlsanity.com/paper/2202.06854/) (2022)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Autoencoding Hyperbolic Representation for Adversarial Generation](https://mlsanity.com/paper/2201.12825/) (2022)
- [PHyCLIP: ℓ1-Product of Hyperbolic Factors Unifies Hierarchy and Compositionality in Vision-Language Representation Learning](https://mlsanity.com/paper/2510.08919/) (2025)
- [Non-linear Embeddings in Hilbert Simplex Geometry](https://mlsanity.com/paper/2203.11434/) (2022)
- [Low-distortion and GPU-compatible Tree Embeddings in Hyperbolic Space](https://mlsanity.com/paper/2502.17130/) (2025)
- [Complex Hyperbolic Knowledge Graph Embeddings with Fast Fourier Transform](https://mlsanity.com/paper/2211.03635/) (2022)
- [Principal Component Analysis in Space Forms](https://mlsanity.com/paper/2301.02750/) (2023)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)
- [Klein Model for Hyperbolic Neural Networks](https://mlsanity.com/paper/2410.16813/) (2024)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [SINCERE: Sequential Interaction Networks representation learning on Co-Evolving RiEmannian manifolds](https://mlsanity.com/paper/2305.03883/) (2023)
- [Capacity Bounds for Hyperbolic Neural Network Representations of Latent Tree Structures](https://mlsanity.com/paper/2308.09250/) (2023)
- [HyperAid: Denoising in hyperbolic spaces for tree-fitting and hierarchical clustering](https://mlsanity.com/paper/2205.09721/) (2022)
- [Three Revisits to Node-Level Graph Anomaly Detection: Outliers, Message Passing and Hyperbolic Neural Networks](https://mlsanity.com/paper/2403.04010/) (2024)
- [Hyperbolic Hierarchical Knowledge Graph Embeddings for Link Prediction in Low Dimensions](https://mlsanity.com/paper/2204.13704/) (2022)
- [Hyperbolic Convolution via Kernel Point Aggregation](https://mlsanity.com/paper/2306.08862/) (2023)
- [Improving Hyperbolic Representations via Gromov-Wasserstein Regularization](https://mlsanity.com/paper/2407.10495/) (2024)
- [Mixed-curvature decision trees and random forests](https://mlsanity.com/paper/2410.13879/) (2024)
- [Hyperbolic Graph Neural Networks Under the Microscope: The Role of Geometry-Task Alignment](https://mlsanity.com/paper/2602.01828/) (2026)
- [Unit Ball Model for Embedding Hierarchical Structures in the Complex Hyperbolic Space](https://mlsanity.com/paper/2105.03966/) (2021)
- [Why are hyperbolic neural networks effective? A study on hierarchical representation capability](https://mlsanity.com/paper/2402.02478/) (2024)
- [Identifying the Hierarchical Emotional Areas in the Human Brain Through Information Fusion](https://mlsanity.com/paper/2408.00525/) (2024)

## Abstract (arXiv)

Given data, finding a faithful low-dimensional hyperbolic embedding of the data is a key method by which we can extract hierarchical information or learn representative geometric features of the data. In this paper, we explore a new method for learning hyperbolic representations by taking a metric-first approach. Rather than determining the low-dimensional hyperbolic embedding directly, we learn a tree structure on the data. This tree structure can then be used directly to extract hierarchical information, embedded into a hyperbolic manifold using Sarkar's construction citesarkar, or used as a tree approximation of the original metric. To this end, we present a novel fast algorithm textscTreeRep such that, given a δ-hyperbolic metric (for any δ ≥ 0), the algorithm learns a tree structure that approximates the original metric. In the case when δ = 0, we show analytically that textscTreeRep exactly recovers the original tree structure. We show empirically that textscTreeRep is not only many orders of magnitude faster than previously known algorithms, but also produces metrics with lower average distortion and higher mean average precision than most previous algorithms for learning hyperbolic embeddings, extracting hierarchical information, and approximating metrics via tree metrics.