ICML 2025

Low-distortion and GPU-compatible Tree Embeddings in Hyperbolic Space
Max van Spengler 1 Pascal Mettes 1
Abstract
Embedding tree-like data, from hierarchies to on-
tologies and taxonomies, forms a well-studied
problem for representing knowledge across many
domains. Hyperbolic geometry provides a natu-
ral solution for embedding trees, with vastly su-
perior performance over Euclidean embeddings.
Recent literature has shown that hyperbolic tree
embeddings can even be placed on top of neu-
ral networks for hierarchical knowledge integra-
tion in deep learning settings. For all applica-
tions, a faithful embedding of trees is needed,
with combinatorial constructions emerging as the
most effective direction. This paper identifies and
solves two key limitations of existing works. First,
the combinatorial construction hinges on finding
highly separated points on a hypersphere, a no-
toriously difficult problem. Current approaches
achieve poor separation, degrading the quality of
the corresponding hyperbolic embedding. We pro-
pose highly separated Delaunay tree embeddings
(HS-DTE), which integrates angular separation
in a generalized formulation of Delaunay embed-
dings, leading to lower embedding distortion. Sec-
ond, low-distortion requires additional precision.
The current approach for increasing precision is
to use multiple precision arithmetic, which ren-
ders the embeddings useless on GPUs in deep
learning settings. We reformulate the combinato-
rial construction using floating point expansion
arithmetic, leading to superior embedding quality
while retaining utility on accelerated hardware.
1. Introduction
Tree-like structures such as hierarchies are key for knowl-
edge representation, from biological taxonomies (Padial
et al., 2010) and phylogenetics (Kapli et al., 2020) to natu-
1VIS Lab, University of Amsterdam, The Netherlands. Corre-
spondence to: Max van Spengler <m.w.f.vanspengler@uva.nl>.
Proceedings of the 42 nd International Conference on Machine
Learning,...