Graph-based Nearest Neighbor Search in Hyperbolic Spaces

The nearest neighbor search (NNS) problem is widely studied in Euclidean space, and graph-based algorithms are known to outperform other approaches for this task. However, hyperbolic geometry is found to be very useful for data representation in various domains, including natural language processing, computer vision, and information retrieval. In this paper, we show that graph-based approaches are also well suited for hyperbolic geometry. From a theoretical perspective, we rigorously analyze the time and space complexity of graph-based NNS, assuming that an n-element dataset is uniformly distributed within a d-dimensional ball of radius R in the hyperbolic space of curvature -1. Assuming the dense setting (d << log(n)), we derive the time and space complexity of graph-based NNS and compare the obtained results with known guarantees for the Euclidean case. Interestingly, under some assumptions on dimension d and radius R, graph-based NNS has lower time complexity in the hyperbolic space. From a practical perspective, we illustrate this result on word embedding data: we compare graph-based NNS for GloVe and Poincare GloVe word embeddings. It turns out that for the same corpus, graph-based NNS is more efficient in the hyperbolic space. We also demonstrate that graph-based methods outperform other existing baselines on Poincare GloVe word embeddings. Overall, our theoretical and empirical analysis suggests that graph-based NNS can be considered a default approach for similarity search in hyperbolic spaces.