Efficient Variants of Wasserstein Distance in Hyperbolic Space via Space-Filling Curve Projection.

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Hyperbolic spaces have been considered pervasively for embedding hierarchically structured data in the recent decade. However, there is a lack of studies focusing on efficient distance metrics for comparing probability distributions in hyperbolic spaces. To bridge the gap, we propose a novel metric called the hyperbolic space-filling curve projection Wasserstein (SFW) distance. The idea is to first project two probability distributions onto a space-filling curve to obtain a closed-form coupling between them and then calculate the transport distance between these two distributions in the hyperbolic space accordingly. Theoretically, we show the SFW distance is a proper metric and is well-defined for probability measures with bounded supports. Statistical convergence rates for the proposed estimator are provided as well. Moreover, we propose two variants of the SFW distance based on geodesic and horospherical projections, respectively, to combat the curse-of-dimensionality. Empirical results on synthetic and real-world data indicate that the SFW distance can effectively serve as a surrogate of the popular Wasserstein distance with low complexity.