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Hyperbolic Delaunay Geometric Alignment

Aniss Aiman Medbouhi, Giovanni Luca Marchetti, Vladislav Polianskii, Alexander Kravberg, Petra Poklukar, Anastasia Varava, Danica Kragic

TL;DR

This work tackles the lack of evaluation tools for hyperbolic embeddings by introducing HyperDGA, a geometry-aware similarity score between two sets embedded in hyperbolic space. HyperDGA is computed from the hyperbolic Delaunay graph via the ratio of heterogeneous to total edges, leveraging the Klein-Beltrami model to reduce hyperbolic computations to Euclidean power diagrams. Through synthetic experiments with a Hyperbolic VAE and real single-cell RNA data embedded in hyperbolic space, HyperDGA demonstrates strong correlation with data perturbations and latent representation quality, often outperforming hyperbolic Chamfer and matching or exceeding hyperbolic Wasserstein. The work provides a practical tool for evaluating and comparing hyperbolic representations and suggests future directions toward differentiable variants and broader biomedical applications.

Abstract

Hyperbolic machine learning is an emerging field aimed at representing data with a hierarchical structure. However, there is a lack of tools for evaluation and analysis of the resulting hyperbolic data representations. To this end, we propose Hyperbolic Delaunay Geometric Alignment (HyperDGA) -- a similarity score for comparing datasets in a hyperbolic space. The core idea is counting the edges of the hyperbolic Delaunay graph connecting datapoints across the given sets. We provide an empirical investigation on synthetic and real-life biological data and demonstrate that HyperDGA outperforms the hyperbolic version of classical distances between sets. Furthermore, we showcase the potential of HyperDGA for evaluating latent representations inferred by a Hyperbolic Variational Auto-Encoder.

Hyperbolic Delaunay Geometric Alignment

TL;DR

This work tackles the lack of evaluation tools for hyperbolic embeddings by introducing HyperDGA, a geometry-aware similarity score between two sets embedded in hyperbolic space. HyperDGA is computed from the hyperbolic Delaunay graph via the ratio of heterogeneous to total edges, leveraging the Klein-Beltrami model to reduce hyperbolic computations to Euclidean power diagrams. Through synthetic experiments with a Hyperbolic VAE and real single-cell RNA data embedded in hyperbolic space, HyperDGA demonstrates strong correlation with data perturbations and latent representation quality, often outperforming hyperbolic Chamfer and matching or exceeding hyperbolic Wasserstein. The work provides a practical tool for evaluating and comparing hyperbolic representations and suggests future directions toward differentiable variants and broader biomedical applications.

Abstract

Hyperbolic machine learning is an emerging field aimed at representing data with a hierarchical structure. However, there is a lack of tools for evaluation and analysis of the resulting hyperbolic data representations. To this end, we propose Hyperbolic Delaunay Geometric Alignment (HyperDGA) -- a similarity score for comparing datasets in a hyperbolic space. The core idea is counting the edges of the hyperbolic Delaunay graph connecting datapoints across the given sets. We provide an empirical investigation on synthetic and real-life biological data and demonstrate that HyperDGA outperforms the hyperbolic version of classical distances between sets. Furthermore, we showcase the potential of HyperDGA for evaluating latent representations inferred by a Hyperbolic Variational Auto-Encoder.
Paper Structure (25 sections, 1 theorem, 9 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 25 sections, 1 theorem, 9 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Hyperbolic Dimensionality Reduction
  4. Hyperbolic VAEs
  5. Evaluation of Euclidean Data Representations
  6. Background
  7. Voronoi Cells and Delaunay Graph
  8. The Klein-Beltrami Model
  9. Method
  10. Conversion to Klein-Beltrami
  11. Hyperbolic Voronoi Diagram in $\mathbb{K}^n$
  12. HyperDGA
  13. Experiments
  14. Synthetic Data with Hyperbolic VAE
  15. Experiment 1: Distance Behavior and Noise Inference.
  16. ...and 10 more sections

Key Result

theorem thmcountertheorem

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 (9)

  • Figure 1: Illustration of HyperDGA for a hyperbolic representation of hierarchical data. Our score counts heterogeneous edges (black) of the Delaunay graph, which is dual to the hyperbolic Voronoi diagram (gray). Data closer in the hierarchy (green & yellow) have lower scores than data further apart (red & green).
  • Figure 2: Depiction of the Euclidean power diagram (gray) equivalent to the hyperbolic Voronoi diagram, the hyperbolic Delaunay graph (green), and the pruned edges (red). Data is obtained from real-life cells (neoblasts 4 & 7, Plass2018_planaria_SingleCellData) via Poincaré embedding Nickel_2020_SingleCellPoincareEmbedding and converted to the Klein-Beltrami model.
  • Figure 3: Left: Klein-Beltrami hyperbolic representation (via Poincaré embedding Nickel_2020_SingleCellPoincareEmbedding) of neoblasts 4 & 7 Plass2018_planaria_SingleCellData (blue and orange). Right: Depiction of their corresponding homogeneous edges (blue and orange) and heterogeneous edges (green).
  • Figure 4: Top: Poincaré hyperbolic encodings of the training set (blue) and $A_{\epsilon}$ (orange). Bottom: Corresponding Klein-Beltrami visualizations of HyperDGA, with homogeneous edges (blue & orange) and heterogeneous ones (green).
  • Figure 5: Hyperbolic distances between the encodings of the training set and $A_{\epsilon}$ in function of the random perturbation $\epsilon$.
  • ...and 4 more figures

Theorems & Definitions (4)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem thmcountertheorem: HyperbolicVoronoiDiagramsMadeEasy2010
  • definition thmcounterdefinition