# HyperSteiner: Computing Heuristic Hyperbolic Steiner Minimal Trees

arXiv: https://arxiv.org/abs/2409.05671 · HTML: https://mlsanity.com/paper/2409.05671/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2024.

## Built on by (most cited first, 3 of 3)

- [Klein Model for Hyperbolic Neural Networks](https://mlsanity.com/paper/2410.16813/) (2024)
- [Randomized HyperSteiner: A Stochastic Delaunay Triangulation Heuristic for the Hyperbolic Steiner Minimal Tree](https://mlsanity.com/paper/2510.09328/) (2025)
- [Even Faster Hyperbolic Random Forests: A Beltrami-Klein Wrapper Approach](https://mlsanity.com/paper/2506.04360/) (2025)

## Abstract (arXiv)

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.