  Geometric representation learning has recently shown great promise in several
machine learning settings, ranging from relational learning to language
processing and generative models. In this work, we consider the problem of
performing manifold-valued regression onto an hyperbolic space as an
intermediate component for a number of relevant machine learning applications.
In particular, by formulating the problem of predicting nodes of a tree as a
manifold regression task in the hyperbolic space, we propose a novel
perspective on two challenging tasks: 1) hierarchical classification via label
embeddings and 2) taxonomy extension of hyperbolic representations. To address
the regression problem we consider previous methods as well as proposing two
novel approaches that are computationally more advantageous: a parametric deep
learning model that is informed by the geodesics of the target space and a
non-parametric kernel-method for which we also prove excess risk bounds. Our
experiments show that the strategy of leveraging the hyperbolic geometry is
promising. In particular, in the taxonomy expansion setting, we find that the
hyperbolic-based estimators significantly outperform methods performing
regression in the ambient Euclidean space.
