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

TL;DR

This paper introduces TreeRep, a fast algorithm for learning tree structures from hyperbolic metrics, enabling efficient hierarchical data representation and superior metric approximation compared to prior methods.

Contribution

The paper presents a novel, efficient algorithm for learning tree structures from hyperbolic metrics, improving speed and accuracy over existing methods.

Findings

TreeRep exactly recovers the original tree when δ=0.

TreeRep is many orders of magnitude faster than previous algorithms.

TreeRep achieves lower average distortion and higher mean average precision.

Abstract

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 \cite{sarkar}, or used as a tree approximation of the original metric. To this end, we present a novel fast algorithm \textsc{TreeRep} such that, given a $\delta$-hyperbolic metric (for any $\delta \geq 0$), the algorithm learns a tree structure that approximates the original metric. In the case when $\delta = 0$, we show analytically that \textsc{TreeRep} exactly recovers the original tree structure. We show empirically that \textsc{TreeRep} 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.