Abstract
Hyperbolic embedding methods are defined as state-of-the-art approaches in machine learning and data science for effectively modelling hierarchical structures. Traditional placement methods, rooted in Euclidean geometry, fall short when it comes to expressing hierarchical relations where exponential growth is inevitable. Existing approaches for hierarchical data modelling, such as taxonomies, knowledge graphs, phylogenetic trees and social networks, are often better modelled in non-Euclidean spaces due to their structural properties. Hyperbolic Geometry is a mathematically sound embedding space that is especially well-suited for minimally distorting high dimensional data due to its inherent negative curvature. Hyperbolic Embedding: A Survey on Theory, Methodology and Applications of Representation for Hierarchical Data Abstract: It starts with the explanation of Euclidean embedding’s and its issues in taking into account multi-level hierarchical relationships, providing hyperbolic space as an alternative solution. It investigates core models, including the Poincare ball model and hyperboloid model, which facilitate capturing tree-like structures as geometric properties. It also covers advanced embedding methods such as Poincare embedding’s and Lorentz embedding’s, and optimization techniques on a Riemannian manifold.
We further explore an integration of hyperbolic embedding’s with contemporary deep learning, from hyperbolic neural networks to graph neural networks. Such techniques help learn hierarchical patterns more effectively in large-sized datasets. Experiments show that hyperbolic embedding’s outperform traditional Euclidean methods in representation quality, scalability, and hierarchical relationships. Besides the theoretical and experimental analysis, this article covers a diverse selection of practical applications such as natural language processing(s), recommender systems, social network analysis, and bioinformatics as relevant to RC. While putting embedding’s in a hyperbolic space have its many benefits, they also come with challenges like numerical stability, high computation complexity and difficulties in visualization/interpreting. The paper ends with highlighting existing research challenges and proposing potential future directions, such as hybrid geometric models and dynamic hierarchical embedding’s. In conclusion, hyperbolic embedding methods are a major improvement from the previous ones and provides a solid foundation for representation learning in any modern machine learning system requiring modelling of complex hierarchical data.
Keywords
Hyperbolic Embedding, Hierarchical Data Representation, Poincare Ball Model, Hyperboloid Model, Riemannian Optimization, Graph Neural Networks, Non-Euclidean Geometry; A Semantic Sense Check On Background Reading
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