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 often allows for better data representation in various domains, including graphs, words, and images. 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 $-1$. Under some conditions on $R$ and $d$, we derive the time and space complexity of graph-based NNS and compare the obtained results with known guarantees for the Euclidean case. Interestingly, in the dense setting ($d \ll \log n$) and under some assumptions on the radius $R$, graph-based NNS has lower time complexity in the hyperbolic space. This agrees with our experiments: we consider datasets embedded in hyperbolic and Euclidean spaces and show that graph-based NNS can be more efficient in the hyperbolic space. We also demonstrate that graph-based methods outperform other existing baselines for hyperbolic NNS. Overall, our theoretical and empirical analysis suggests that graph-based NNS can be considered a default approach for similarity search in hyperbolic spaces.