  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.
