Continuous Hierarchical Representations with Poincaré Variational Auto-Encoders

The variational auto-encoder (VAE) is a popular method for learning a
generative model and embeddings of the data. Many real datasets are
hierarchically structured. However, traditional VAEs map data in a Euclidean
latent space which cannot efficiently embed tree-like structures. Hyperbolic
spaces with negative curvature can. We therefore endow VAEs with a Poincar\'e
ball model of hyperbolic geometry as a latent space and rigorously derive the
necessary methods to work with two main Gaussian generalisations on that space.
We empirically show better generalisation to unseen data than the Euclidean
counterpart, and can qualitatively and quantitatively better recover
hierarchical structures.