# Hyperbolic Distance Matrices

arXiv: https://arxiv.org/abs/2005.08672 · HTML: https://mlsanity.com/paper/2005.08672/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2020.

## Built on by (most cited first, 7 of 7)

- [Generalization Error Bound for Hyperbolic Ordinal Embedding](https://mlsanity.com/paper/2105.10475/) (2021)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [Linear Classifiers in Product Space Forms](https://mlsanity.com/paper/2102.10204/) (2021)
- [Principal Component Analysis in Space Forms](https://mlsanity.com/paper/2301.02750/) (2023)
- [Strain-Minimizing Hyperbolic Network Embeddings with Landmarks](https://mlsanity.com/paper/2207.06775/) (2022)
- [Why are hyperbolic neural networks effective? A study on hierarchical representation capability](https://mlsanity.com/paper/2402.02478/) (2024)
- [Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection](https://mlsanity.com/paper/2512.18826/) (2025)

## Abstract (arXiv)

Hyperbolic space is a natural setting for mining and visualizing data with hierarchical structure. In order to compute a hyperbolic embedding from comparison or similarity information, one has to solve a hyperbolic distance geometry problem. In this paper, we propose a unified framework to compute hyperbolic embeddings from an arbitrary mix of noisy metric and non-metric data. Our algorithms are based on semidefinite programming and the notion of a hyperbolic distance matrix, in many ways parallel to its famous Euclidean counterpart. A central ingredient we put forward is a semidefinite characterization of the hyperbolic Gramian -- a matrix of Lorentzian inner products. This characterization allows us to formulate a semidefinite relaxation to efficiently compute hyperbolic embeddings in two stages: first, we complete and denoise the observed hyperbolic distance matrix; second, we propose a spectral factorization method to estimate the embedded points from the hyperbolic distance matrix. We show through numerical experiments how the flexibility to mix metric and non-metric constraints allows us to efficiently compute embeddings from arbitrary data.