  Finding meaningful representations and distances of hierarchical data is
important in many fields. This paper presents a new method for hierarchical
data embedding and distance. Our method relies on combining diffusion geometry,
a central approach to manifold learning, and hyperbolic geometry. Specifically,
using diffusion geometry, we build multi-scale densities on the data, aimed to
reveal their hierarchical structure, and then embed them into a product of
hyperbolic spaces. We show theoretically that our embedding and distance
recover the underlying hierarchical structure. In addition, we demonstrate the
efficacy of the proposed method and its advantages compared to existing methods
on graph embedding benchmarks and hierarchical datasets.
