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HyperSteiner: Computing Heuristic Hyperbolic Steiner Minimal Trees

Alejandro García-Castellanos, Aniss Aiman Medbouhi, Giovanni Luca Marchetti, Erik J. Bekkers, Danica Kragic

TL;DR

HyperSteiner addresses the SMT problem in hyperbolic space by extending the Euclidean Smith-Lee-Liebman approach to the Klein-Beltrami model. It builds a hyperbolic Delaunay framework, computes Fermat (Steiner) points via isoptic-curve formulations, and assembles local solutions into a global heuristic SMT with complexity $O(|P|\log|P|)$. Empirically, HyperSteiner yields more realistic hierarchies than MST and scales better than Neighbor Joining on large datasets, with Planaria cell data illustrating practical hierarchy discovery benefits. The method enables efficient hyperbolic SMTs suitable for data with intrinsic tree-like structure, offering a scalable tool for hierarchy inference and data analysis in hyperbolic latent spaces.

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.

HyperSteiner: Computing Heuristic Hyperbolic Steiner Minimal Trees

TL;DR

HyperSteiner addresses the SMT problem in hyperbolic space by extending the Euclidean Smith-Lee-Liebman approach to the Klein-Beltrami model. It builds a hyperbolic Delaunay framework, computes Fermat (Steiner) points via isoptic-curve formulations, and assembles local solutions into a global heuristic SMT with complexity . Empirically, HyperSteiner yields more realistic hierarchies than MST and scales better than Neighbor Joining on large datasets, with Planaria cell data illustrating practical hierarchy discovery benefits. The method enables efficient hyperbolic SMTs suitable for data with intrinsic tree-like structure, offering a scalable tool for hierarchy inference and data analysis in hyperbolic latent spaces.

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.
Paper Structure (27 sections, 7 theorems, 16 equations, 7 figures, 7 tables, 1 algorithm)

This paper contains 27 sections, 7 theorems, 16 equations, 7 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Polynomial-Time Approximations of SMTs.
  4. Non-Euclidean SMTs.
  5. Hyperbolic Machine Learning.
  6. Hierarchy Discovery.
  7. Background
  8. Steiner Minimal Trees.
  9. The SLL Algorithm.
  10. The Klein-Beltrami Model.
  11. Method
  12. Hyperbolic Delaunay Triangulation.
  13. Fermat Points.
  14. 4-Terminal Case.
  15. Approximation Error and Computational Complexity.
  16. ...and 12 more sections

Key Result

Theorem 4.1

Given $P \subseteq \mathbb{K}^n$, there exists an explicit set $S \subseteq \mathbb{R}^n$ and weights $\{ r_s\}_{s \in S}$ such that the hyperbolic Voronoi cells of $P$ correspond to restrictions to $\mathbb{K}^n$ of power cells of $S$.

Figures (7)

  • Figure 1: A hyperbolic Steiner tree computed via HyperSteiner. Red ($\newmoon$) denotes terminals, blue ($\newmoon$) denotes Steiner points, while the dashed line corresponds to the auxiliary hyperbolic Delaunay triangulation.
  • Figure 2: Example of isoptic curves for $\alpha=2\pi/3$. Each color represents a different curve. Square ($\,\blacksquare\,$) denotes terminals while star ($\bigstar$) denotes Steiner points.
  • Figure 3: Example of the algebraic curves $\psi$. The solid line corresponds to the isoptic curve for $\alpha=2\pi/3$, and the dashed line to the one for the complementary angle.
  • Figure 4: Illustration of the datasets considered. Left and center: $|P|=1000$ random samples from the synthetic datasets. Right: $|P| = 20000$ samples from the real-life dataset, where less saturated color represents older cells.
  • Figure 5: Convergence dataset for $d=4$. Orange crosses represent the means of $\mathcal{G}(\mu_{4, k}(t), 0.15)$.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4.1: HyperbolicVoronoiDiagramsMadeEasy2010
  • Theorem 4.2: Weng2001SteinerTreesCurvedSurfaces
  • Definition 4
  • Proposition 4.3
  • Proposition 4.4
  • Theorem 4.5: halverson2005steiner
  • Proposition A .1: xu_introduction_2021
  • ...and 1 more