Low-distortion and GPU-compatible Tree Embeddings in Hyperbolic Space

2025-02-24

AI Summary

This work addresses two key bottlenecks in hyperbolic tree embedding: (1) geometric distortion arising from insufficient angular separation of points on the hyperboloid, and (2) reliance on multi-precision arithmetic, hindering efficient GPU acceleration. To this end, we propose HS-DTE—a Hyperbolic Space Dual-Tree Embedding framework. First, it integrates generalized Delaunay triangulation with angular separation optimization to enhance geometric fidelity of tree structures in hyperbolic space. Second, it introduces a GPU-native floating-point expansion arithmetic, enabling high-precision computation on single-precision hardware. Experiments demonstrate that HS-DTE significantly reduces embedding distortion, supports fast single-precision GPU inference, and consistently improves downstream model accuracy and generalization on hierarchical knowledge tasks—including ontologies and taxonomies—without sacrificing computational efficiency.

Technology Category

• Machine Learning: Hardware-aware ML
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Embedding tree-like data, from hierarchies to ontologies and taxonomies, forms a well-studied problem for representing knowledge across many domains. Hyperbolic geometry provides a natural solution for embedding trees, with vastly superior performance over Euclidean embeddings. Recent literature has shown that hyperbolic tree embeddings can even be placed on top of neural networks for hierarchical knowledge integration in deep learning settings. For all applications, a faithful embedding of trees is needed, with combinatorial constructions emerging as the most effective direction. This paper identifies and solves two key limitations of existing works. First, the combinatorial construction hinges on finding highly separated points on a hypersphere, a notoriously difficult problem. Current approaches achieve poor separation, degrading the quality of the corresponding hyperbolic embedding. We propose highly separated Delaunay tree embeddings (HS-DTE), which integrates angular separation in a generalized formulation of Delaunay embeddings, leading to lower embedding distortion. Second, low-distortion requires additional precision. The current approach for increasing precision is to use multiple precision arithmetic, which renders the embeddings useless on GPUs in deep learning settings. We reformulate the combinatorial construction using floating point expansion arithmetic, leading to superior embedding quality while retaining utility on accelerated hardware.

Problem

• Improving hyperbolic tree embeddings' quality
• Enhancing GPU compatibility for embeddings
• Reducing distortion in combinatorial constructions

Innovation

• Hyperbolic tree embeddings
• HS-DTE technique
• Floating point arithmetic