**Visualization of Poincaré Embeddings**

M. Nickel and D. Kiela proposed Poincaré Embeddings for learning hierarchical representations (NIPS’17).

One of their tasks consists in embedding words in the Poincaré ball as to preserve entailment links.

Consider a symbolic dataset of words and directed edges between them, indicating that is a subconcept of . Optimize the following

**Training loss**

denotes word embeddings parameters,

denotes observed entailment links ,

is the set of negative parents of .

*In order to better understand the training dynamics induced by this model, we decided to visualize it in 2D, for the mammal subtree of the WordNet hierarchy.
*

**Hyperbolic case**

Using the Poincaré disk distance for , and optimizing parameters using Riemannian SGD, for embeddings in dimension 2, yields the following dynamics:

**Euclidean case**

In comparison, using the Euclidean distance for and optimizing parameters using SGD, for embeddings in dimension 2, yields the following dynamics:

In both simulations, initialization was done with a burn-in phase, i.e. with small learning rate, as suggested by Nickel and Kiela.