  Recently, there has been a surge of interest in representation learning in
hyperbolic spaces, driven by their ability to represent hierarchical data with
significantly fewer dimensions than standard Euclidean spaces. However, the
viability and benefits of hyperbolic spaces for downstream machine learning
tasks have received less attention. In this paper, we present, to our
knowledge, the first theoretical guarantees for learning a classifier in
hyperbolic rather than Euclidean space. Specifically, we consider the problem
of learning a large-margin classifier for data possessing a hierarchical
structure. We provide an algorithm to efficiently learn a large-margin
hyperplane, relying on the careful injection of adversarial examples. Finally,
we prove that for hierarchical data that embeds well into hyperbolic space, the
low embedding dimension ensures superior guarantees when learning the
classifier directly in hyperbolic space.
