The hyperbolic manifold is a smooth manifold of negative constant curvature. While the hyperbolic manifold is well-studied in the literature, it has gained interest in the machine learning and natural language processing communities lately due to its usefulness in modeling continuous hierarchies. Tasks with hierarchical structures are ubiquitous in those fields and there is a general interest to learning hyperbolic representations or embeddings of such tasks. Additionally, these embeddings of related tasks may also share a low-rank subspace. In this work, we propose to learn hyperbolic embeddings such that they also lie in a low-dimensional subspace. In particular, we consider the problem of learning a low-rank factorization of hyperbolic embeddings. We cast these problems as manifold optimization problems and propose computationally efficient algorithms. Empirical results illustrate the efficacy of the proposed approach.
Embeddings of tree-like graphs in hyperbolic space were recently shown to surpass their Euclidean counterparts in performance by a large margin. Inspired by these results, we present an algorithm for learning word embeddings in hyperbolic space from free text. An objective function based on the hyperbolic distance is derived and included in the skip-gram architecture from word2vec. The hyperbolic word embeddings are then evaluated on word similarity and analogy benchmarks. The results demonstrate the potential of hyperbolic word embeddings, particularly in low dimensions, though without clear superiority over their Euclidean counterparts. We further discuss problems in the formulation of the analogy task resulting from the curvature of hyperbolic space.
We present a large scale hyperbolic recommender system. We discuss why hyperbolic geometry is a more suitable underlying geometry for many recommendation systems and cover the fundamental milestones and insights that we have gained from its development. In doing so, we demonstrate the viability of hyperbolic geometry for recommender systems, showing that they significantly outperform Euclidean models on datasets with the properties of complex networks. Key to the success of our approach are the novel choice of underlying hyperbolic model and the use of the Einstein midpoint to define an asymmetric recommender system in hyperbolic space. These choices allow us to scale to millions of users and hundreds of thousands of items.
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCN operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the Pubmed dataset.
Neural embeddings have been used with great success in Natural Language Processing (NLP). They provide compact representations that encapsulate word similarity and attain state-of-the-art performance in a range of linguistic tasks. The success of neural embeddings has prompted significant amounts of research into applications in domains other than language. One such domain is graph-structured data, where embeddings of vertices can be learned that encapsulate vertex similarity and improve performance on tasks including edge prediction and vertex labelling. For both NLP and graph based tasks, embeddings have been learned in high-dimensional Euclidean spaces. However, recent work has shown that the appropriate isometric space for embedding complex networks is not the flat Euclidean space, but negatively curved, hyperbolic space. We present a new concept that exploits these recent insights and propose learning neural embeddings of graphs in hyperbolic space. We provide experimental evidence that embedding graphs in their natural geometry significantly improves performance on downstream tasks for several real-world public datasets.