Poincaré Embeddings for Learning Hierarchical Representations (1705.08039v2)

Abstract: Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this property. For this purpose, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional Poincar\'e ball. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity. We introduce an efficient algorithm to learn the embeddings based on Riemannian optimization and show experimentally that Poincar\'e embeddings outperform Euclidean embeddings significantly on data with latent hierarchies, both in terms of representation capacity and in terms of generalization ability.

• The paper presents Poincaré embeddings as a novel method for capturing hierarchical structures by mapping symbolic data into hyperbolic space.
• It employs efficient Riemannian optimization with negative sampling to enable scalable training on large, complex datasets.
• Empirical evaluations demonstrate superior performance in tasks like taxonomy embedding and lexical entailment compared to Euclidean methods.

Poincaré Embeddings for Learning Hierarchical Representations: An Overview

The paper "Poincaré Embeddings for Learning Hierarchical Representations" by Maximilian Nickel and Douwe Kiela represents a significant advancement in the field of representation learning for hierarchical data. The authors introduce an innovative approach for embedding data into hyperbolic space, specifically an $n$-dimensional Poincaré ball, in order to capture both the hierarchy and the similarity inherent in symbolic datasets.

Context and Motivation

Representation learning has become a cornerstone in machine learning, particularly for text and graph data. Traditional methods such as word2vec, GloVe, and FastText have been extensively used to capture semantic relationships in Euclidean spaces. However, these approaches do not efficiently represent the latent hierarchical structures often present in complex datasets. The authors address this limitation by leveraging hyperbolic geometry, which naturally accommodates hierarchical data more effectively than Euclidean spaces.

Methodological Contributions

The core contribution of this paper is the introduction of Poincaré embeddings, which map symbolic data into a hyperbolic space, represented as an $n$-dimensional Poincaré ball. The unique properties of hyperbolic space, such as its exponential growth of distances from the origin, make it well-suited for representing hierarchical structures.

A critical aspect of the proposed method is the efficient algorithm for learning these embeddings based on Riemannian optimization. The authors detail the mathematical framework for this optimization, including the derivation of the Riemannian gradient and the choice of suitable retraction operations. The resulting updates allow for scalable training even on large datasets, thanks to a natural gradient approach and efficient negative sampling.

Empirical Evaluation

The effectiveness of Poincaré embeddings is demonstrated through extensive empirical evaluations. The authors evaluate their method on several tasks:

1. Embedding Taxonomies: Using the transitive closure of the WordNet noun hierarchy, the authors show that Poincaré embeddings significantly reduce the mean rank and increase mean average precision (MAP) compared to Euclidean and translational embeddings. The embeddings also prove robust across different dimensions, highlighting their capacity for embedding large taxonomies.
2. Network Embeddings: For social networks like AstroPh, CondMat, GrQc, and HepPh, Poincaré embeddings outperform Euclidean embeddings, particularly in low-dimensional settings. The use of the Fermi-Dirac distribution to model edge probabilities further underscores the suitability of hyperbolic space for capturing latent hierarchical structures in complex networks.
3. Lexical Entailment: By leveraging the hierarchical nature of Poincaré embeddings, the authors achieve state-of-the-art performance on the HyperLex dataset for graded lexical entailment. The embeddings are also effective for non-graded lexical entailment as indicated by their performance on the Wbless dataset.

Theoretical and Practical Implications

The introduction of Poincaré embeddings has several important implications:

• Theoretical: This work establishes hyperbolic space as a powerful alternative to Euclidean space for representation learning, specifically for hierarchical data. It offers a new perspective on modeling complex relationships that are not easily captured by traditional methods.
• Practical: The proposed method provides a scalable and efficient solution for embedding large and complex datasets. The embeddings' ability to simultaneously capture hierarchy and similarity has practical applications in various domains, including natural language processing, social network analysis, and knowledge graph completion.

Future Directions

The authors hint at several future directions for this line of research. One promising avenue is the extension of Poincaré embeddings to multi-relational data, which could further enhance the embeddings’ applicability and utility. Additionally, further exploration into specialized models for specific applications, such as word embeddings, might yield even more precise and efficient representations. Finally, adopting a full Riemannian optimization approach could improve the quality and convergence speed of the embeddings.

In summary, the paper "Poincaré Embeddings for Learning Hierarchical Representations" makes a compelling case for the utility of hyperbolic geometry in embedding hierarchical data, provides a robust algorithm for computing such embeddings, and demonstrates their efficacy across a range of benchmark tasks. The work lays a strong foundation for future research and application in this burgeoning field.