Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Achieving Hyperbolic-Like Expressiveness with Arbitrary Euclidean Regions: A New Approach to Hierarchical Embeddings

Hui Yang, Jiaoyan Chen

TL;DR

The paper addresses the challenge of embedding hierarchical data in low dimensions using hyperbolic geometry by introducing RegD, a Euclidean framework that embeds arbitrary regions (e.g., boxes, balls). RegD combines two dissimilarities, depth $d_{\text{dep}}$ and boundary $d_{\text{bd}}$, to mimic hyperbolic expressiveness and support richer semantic relations such as ontology reasoning. The authors establish theoretical connections showing $d_{\text{dep}}$ recovers hyperbolic distance as a special case and demonstrate strong empirical gains on DAGs and ontologies over state-of-the-art baselines, while maintaining simplicity and flexibility. This approach offers a versatile tool for hierarchical and semantic embedding tasks, with reproducibility resources and potential for seamless integration with hyperbolic neural networks and broader region types.

Abstract

Hierarchical data is common in many domains like life sciences and e-commerce, and its embeddings often play a critical role. While hyperbolic embeddings offer a theoretically grounded approach to representing hierarchies in low-dimensional spaces, current methods often rely on specific geometric constructs as embedding candidates. This reliance limits their generalizability and makes it difficult to integrate with techniques that model semantic relationships beyond pure hierarchies, such as ontology embeddings. In this paper, we present RegD, a flexible Euclidean framework that supports the use of arbitrary geometric regions -- such as boxes and balls -- as embedding representations. Although RegD operates entirely in Euclidean space, we formally prove that it achieves hyperbolic-like expressiveness by incorporating a depth-based dissimilarity between regions, enabling it to emulate key properties of hyperbolic geometry, including exponential growth. Our empirical evaluation on diverse real-world datasets shows consistent performance gains over state-of-the-art methods and demonstrates RegD's potential for broader applications such as the ontology embedding task that goes beyond hierarchy.

Achieving Hyperbolic-Like Expressiveness with Arbitrary Euclidean Regions: A New Approach to Hierarchical Embeddings

TL;DR

The paper addresses the challenge of embedding hierarchical data in low dimensions using hyperbolic geometry by introducing RegD, a Euclidean framework that embeds arbitrary regions (e.g., boxes, balls). RegD combines two dissimilarities, depth and boundary , to mimic hyperbolic expressiveness and support richer semantic relations such as ontology reasoning. The authors establish theoretical connections showing recovers hyperbolic distance as a special case and demonstrate strong empirical gains on DAGs and ontologies over state-of-the-art baselines, while maintaining simplicity and flexibility. This approach offers a versatile tool for hierarchical and semantic embedding tasks, with reproducibility resources and potential for seamless integration with hyperbolic neural networks and broader region types.

Abstract

Hierarchical data is common in many domains like life sciences and e-commerce, and its embeddings often play a critical role. While hyperbolic embeddings offer a theoretically grounded approach to representing hierarchies in low-dimensional spaces, current methods often rely on specific geometric constructs as embedding candidates. This reliance limits their generalizability and makes it difficult to integrate with techniques that model semantic relationships beyond pure hierarchies, such as ontology embeddings. In this paper, we present RegD, a flexible Euclidean framework that supports the use of arbitrary geometric regions -- such as boxes and balls -- as embedding representations. Although RegD operates entirely in Euclidean space, we formally prove that it achieves hyperbolic-like expressiveness by incorporating a depth-based dissimilarity between regions, enabling it to emulate key properties of hyperbolic geometry, including exponential growth. Our empirical evaluation on diverse real-world datasets shows consistent performance gains over state-of-the-art methods and demonstrates RegD's potential for broader applications such as the ontology embedding task that goes beyond hierarchy.

Paper Structure

This paper contains 40 sections, 5 theorems, 33 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Works
  3. Manifold and Hyperbolic Space
  4. Region-based Methods
  5. Hyperbolic Methods
  6. Method
  7. Depth Dissimilarity: A Similarity Measure Incorporating Depth
  8. Construction
  9. Comparison with the Hyperbolic Distance
  10. Boundary Dissimilarity: A Non-Symmetric Measure of Inclusion
  11. Construction
  12. Specific Constructions for Boxes or Balls
  13. Training
  14. Evaluation
  15. Inferences over DAG
  16. ...and 25 more sections

Key Result

Theorem 1

Consider the region space $\mathcal{B}^n$ consisting of balls in $\mathbb{R}\xspace^n$, with the depth dissimilarity defined in Example exp:dist_depth_ball. The following properties hold: The same conclusions hold for boxes, where $r_0 \in \mathbb{R}_{> 0}$ is replaced with $\mathbf{o}\xspace_0 \in (\mathbb{R}\xspace_{>0})^n$, and the condition $r' \leq r_0$ is replaced by $\mathbf{o}\xspace' \le

Figures (5)

  • Figure 1: Demonstration of a taxonomy (left), its embeddings in the hyperbolic space (middle) and in the Euclidean space as boxes (right).
  • Figure 2: Illustration of $d_{\text{bd}}(\textit{reg}_1,\textit{reg}_2)$ (red) for $\textit{reg}_2 \subseteq \textit{reg}_1$ (left) or $\textit{reg}_2 \not\subseteq \textit{reg}_1$ (right). Green lines shows the inverse: $d_{\text{bd}}(\textit{reg}_2,\textit{reg}_1)$.
  • Figure 3: Illustration of internally tangent of item 1 (left) and item 2 (right) in Proposition \ref{['prop:depth_hyper']}.
  • Figure 4: Mapping from balls to cones in the case of dimension 1.
  • Figure 5: All F1 scores of 10 random runs on Mammal, WordNet noun, MCG, and Hearst.

Theorems & Definitions (17)

  • Definition 1: Depth Dissimilarity
  • Example 1
  • Example 2
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Definition 2: Boundary Dissimilarity
  • Example 3
  • Proposition 3
  • proof : Proof of Theorem \ref{['theo:dep_dist']}
  • ...and 7 more