HyperSteiner: Computing Heuristic Hyperbolic Steiner Minimal Trees

TL;DR

HyperSteiner is a new heuristic algorithm for efficiently computing Steiner minimal trees in hyperbolic space, leveraging hyperbolic geometry for better hierarchy inference in large datasets.

Contribution

It extends Euclidean Steiner tree algorithms to hyperbolic space using a novel equation system in the Klein-Beltrami model, enabling scalable hierarchy discovery.

Findings

HyperSteiner infers more realistic hierarchies than MST.

It is more scalable to large datasets than Neighbor Joining.

The method effectively captures hierarchical structures in data.

Abstract

We propose HyperSteiner -- an efficient heuristic algorithm for computing Steiner minimal trees in the hyperbolic space. HyperSteiner extends the Euclidean Smith-Lee-Liebman algorithm, which is grounded in a divide-and-conquer approach involving the Delaunay triangulation. The central idea is rephrasing Steiner tree problems with three terminals as a system of equations in the Klein-Beltrami model. Motivated by the fact that hyperbolic geometry is well-suited for representing hierarchies, we explore applications to hierarchy discovery in data. Results show that HyperSteiner infers more realistic hierarchies than the Minimum Spanning Tree and is more scalable to large datasets than Neighbor Joining.