Representation Tradeoffs for Hyperbolic Embeddings (1804.03329v2)

Abstract: Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

• The paper introduces novel combinatorial methods to embed trees in hyperbolic space, achieving a MAP of 0.989 with just two dimensions on WordNet.
• It quantitatively analyzes tradeoffs between precision and dimensionality, demonstrating efficient low-distortion representations for hierarchical data.
• The research develops hyperbolic multidimensional scaling (h-MDS) and a scalable PyTorch implementation to optimize embeddings for practical applications.

Insights into "Representation Tradeoffs for Hyperbolic Embeddings"

The paper entitled "Representation Tradeoffs for Hyperbolic Embeddings" presents a detailed exploration into the efficiency and constraints of hyperbolic embeddings for hierarchical data. This research is particularly relevant for embedding structures such as synonym hierarchies and taxonomies into lower-dimensional spaces while preserving their essential properties. The authors provide empirical and theoretical insights into the precision-dimensionality tradeoff within hyperbolic embeddings and propose novel algorithms and analyses.

Combating Distortion with Hyperbolic Embeddings

The paper starts by addressing the high efficiency of hyperbolic spaces over Euclidean spaces when dealing with tree-like hierarchical data. The authors propose a combinatorial construction method to embed trees into hyperbolic spaces. On WordNet, a combinatorial embedding achieves a remarkable mean average precision (MAP) of 0.989 using just two dimensions, surpassing previously known approaches that required 200 dimensions to reach a MAP of 0.87.

The authors ground their approach by extending algorithms that facilitate near-perfect embeddings for hierarchical data into hyperbolic space. They leverage the Poincaré disk, a two-dimensional model, to express trees with minimal distortion and subsequently generalize their construction to multi-dimensional scenarios.

Balancing Precision and Quality

A significant contribution of this work is the quantitative analysis of tradeoffs related to embedding precision. The researchers identify dimensionality and data path lengths as critical parameters influencing precision requirements. Their findings suggest that while high-fidelity hyperbolic embeddings offer exponential space advantages for compact hierarchies, extended chains or high precision requirements pose challenges, demanding more resources.

Optimizing Hyperbolic Multidimensional Scaling

To extend the benefits of hyperbolic embeddings to broader applications, the paper introduces hyperbolic multidimensional scaling (h-MDS). This variant of multidimensional scaling is tailored for hyperbolic spaces and demonstrates low distortion in embeddings across various datasets. The authors show that solution accuracy is heavily contingent on the geometric centering of the embedded points, and they propose a novel pseudo-Euclidean mean as a solution.

Learning and Implementation in Practice

The complexity of embedding optimization is addressed through a PyTorch-based stochastic gradient descent implementation. This approach enhances scalability and deals with incomplete distance information, proving robust in real-world applications. The refined algorithm, accommodating noise and learning a scaling factor dynamically, significantly improves MAP scores, particularly when tuning for specific datasets.

Future Directions and Implications

The paper identifies several paths for potential advancement. Foremost among these is the integration of hyperbolic embeddings into machine learning pipelines more effectively. Given their robustness in preserving hierarchical relationships, hyperbolic embeddings could greatly enhance tasks such as natural language processing, information retrieval, and network analysis. Another avenue is refining the interplay between precision, dimensionality, and embedding quality, potentially leading to more resource-efficient algorithms.

In conclusion, this paper elucidates critical aspects of hyperbolic embeddings, emphasizing the balance between precision and dimensionality. The proposed combinatorial constructions, alongside the development of h-MDS, establish essential frameworks for future exploration in embedding technologies. The results indicate that embedding hierarchical data into hyperbolic spaces is not merely a geometric exercise but a practical solution that aligns well with the performance constraints of modern computational applications.