# Hydra: A method for strain-minimizing hyperbolic embedding of network- and distance-based data

## TL;DR

Hydra is a new hyperbolic embedding method that minimizes hyperbolic strain, recovers points exactly on hyperbolic submanifolds, and offers faster computation with competitive or superior embedding quality.

## Contribution

Hydra introduces a mathematically optimal hyperbolic embedding technique with an extended version, hydra+, that improves speed and quality over existing methods.

## Key findings

- Hydra achieves competitive embedding quality with reduced computation time.
- Hydra+ outperforms existing methods in both speed and embedding accuracy.
- The method guarantees exact recovery for points on hyperbolic submanifolds.

## Abstract

We introduce hydra (hyperbolic distance recovery and approximation), a new method for embedding network- or distance-based data into hyperbolic space. We show mathematically that hydra satisfies a certain optimality guarantee: It minimizes the `hyperbolic strain' between original and embedded data points. Moreover, it recovers points exactly, when they are located on a hyperbolic submanifold of the feature space. Testing on real network data we show that the embedding quality of hydra is competitive with existing hyperbolic embedding methods, but achieved at substantially shorter computation time. An extended method, termed hydra+, outperforms existing methods in both computation time and embedding quality.