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

arXiv2005.03847MaRDI QIDQ6340284

A. Gilbert, Author name not available (Why is that?)

Publication date: 8 May 2020

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 extsc{TreeRep} such that, given a $delta$-hyperbolic metric (for any $deltageq0$), the algorithm learns a tree structure that approximates the original metric. In the case when $delta=0$, we show analytically that extsc{TreeRep} exactly recovers the original tree structure. We show empirically that extsc{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.

Has companion code repository: https://github.com/rsonthal/TreeRep

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