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

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 that takes a metric-first approach. Rather than determining the low-dimensional hyperbolic embedding directly, we learn a tree structure on the data as an intermediate step. This tree structure can then be used directly to extract hierarchical information, embedded into a hyperbolic manifold using Sarkar's construction (Sarkar, 2012), or used as a tree approximation of the original metric. To this end, we present a novel fast algorithm 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 TreeRep exactly recovers the original tree structure. We show empirically that TreeRep is not only many orders of magnitude faster than previous 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.