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

Area of Science:

• Machine Learning
• Geometric Deep Learning
• Probability Theory

Background:

• Hyperbolic spaces are increasingly used for embedding hierarchical data.
• Existing methods lack efficient distance metrics for probability distributions in hyperbolic spaces.

Purpose of the Study:

• To propose a novel, efficient distance metric for comparing probability distributions in hyperbolic spaces.
• To address the limitations of current metrics in handling complex, hierarchical data structures.

Main Methods:

• Developed the hyperbolic space-filling curve projection Wasserstein (SFW) distance.
• Utilized space-filling curves for closed-form coupling of probability distributions.
• Analyzed theoretical properties, including metric definition and convergence rates.
• Proposed variants using geodesic and horospherical projections to mitigate the curse-of-dimensionality.

Main Results:

• The SFW distance is theoretically proven to be a proper metric for probability measures with bounded supports.
• Statistical convergence rates for the SFW distance estimator were established.
• Empirical evaluations demonstrated the SFW distance's effectiveness as a low-complexity surrogate for Wasserstein distance.
• Variants showed improved performance in high-dimensional scenarios.

Conclusions:

• The proposed SFW distance provides an efficient and theoretically sound method for comparing probability distributions in hyperbolic spaces.
• This metric offers a viable, computationally less expensive alternative to the Wasserstein distance for applications involving hierarchical data.
• The SFW distance and its variants hold promise for advancing machine learning tasks that leverage hyperbolic geometry.