  Words are not created equal. In fact, they form an aristocratic graph with a
latent hierarchical structure that the next generation of unsupervised learned
word embeddings should reveal. In this paper, justified by the notion of
delta-hyperbolicity or tree-likeliness of a space, we propose to embed words in
a Cartesian product of hyperbolic spaces which we theoretically connect to the
Gaussian word embeddings and their Fisher geometry. This connection allows us
to introduce a novel principled hypernymy score for word embeddings. Moreover,
we adapt the well-known Glove algorithm to learn unsupervised word embeddings
in this type of Riemannian manifolds. We further explain how to solve the
analogy task using the Riemannian parallel transport that generalizes vector
arithmetics to this new type of geometry. Empirically, based on extensive
experiments, we prove that our embeddings, trained unsupervised, are the first
to simultaneously outperform strong and popular baselines on the tasks of
similarity, analogy and hypernymy detection. In particular, for word hypernymy,
we obtain new state-of-the-art on fully unsupervised WBLESS classification
accuracy.
