I gave a talk last night at the Berlin machine learning meetup on learning graph embeddings in hyperbolic space, featuring the recent NIPS 2017 paper of Nickel & Kiela. Covered are:

An illustration of why the Euclidean plane is not a good place to embed trees (since circle circumference grows only linearly in the radius);

Extending this same argument to higher dimensional Euclidean space;

An introduction to the hyperbolic plane and the Poincaré disc model;

A discussion of Rik Sarkar’s result that trees embed with arbitrarily small error in the hyperbolic plane;

A demonstration that, in the hyperbolic plane, circle circumference is exponential in the radius (better written here);

A review of the results of Nickel & Kiela on the (transitive closure of the) WordNet hypernymy graph;

Some thoughts on the gradient optimisation (perhaps better written here).
And here are the slides!