Compact completion and uniform statistical bounds for hyperbolic geodesic sliced Wasserstein distance
Year: 2026
Type: preprint
Abstract: Unrefereed research candidate. Data such as trees, hierarchies and networks are often embedded in hyperbolic space. Geodesic hyperbolic sliced Wasserstein distance (GHSW; Bonet et al., 2023) compares two such data clouds cheaply by projecting them onto geodesics through a fixed centre and averaging one-dimensional transport costs. This paper determines exactly what that distance measures. Results (proved). In the Klein ball each geodesic projection is atanh(v·z), and these projections have uniformly bounded angular moments even at the ideal boundary. Hence, for every dimension d ≥ 2 and 1 ≤ p < ∞, GHSW_p extends to a finite metric on all probability measures with no moment assumption; its metric completion is the space of probability measures on the compact closed Klein ball, it metrises weak convergence there, and its exact diameter is 2C_{d,p}^{1/p}. It therefore cannot control hyperbolic transport moments and is exponentially insensitive to depth far from the centre (exact radial rates e^{-(d-1)r}, r e^{-2pr}, e^{-2pr} across d-1 versus 2p). The induced metric on the boundary sphere is computed in every dimension: squared distance πα for d = 2, (1/2)α^2 log(1/α) behaviour for d = 3, and comparable to α^2 with an exact constant for d ≥ 4. An explicit function shows that the vanishing-at-infinity assertion in Proposition 19 of Bonet, Drumetz and Courty (JMLR 2025) fails. Sampling error obeys distribution-free n^{-1/2} bounds for the powered loss, in expectation and with high probability, with matching minimax rates (n^{-1/(2p)} for the distance itself). Adding one radial Wasserstein term gives a complete metric with the usual hyperbolic Wasserstein topology. Credit and scope: the separation mechanism is due to Lyons (2014) and contraction under geodesic projection is known (Chami et al. 2021). Results concern the exact average over all directions with a fixed origin; finitely many directions never separate laws; minimax constants are not sharp (gap about 20–80); the horospherical construction and general Cartan–Hadamard manifolds remain open. Archive: Pandoc Markdown source, generated LaTeX and PDF, bibliography generated from saved Crossref, arXiv and DataCite records, numerical sanity-check scripts with outputs (normal and python -O), figures, CI and SHA-256 manifest. Assurance: internal AI-assisted review only (not human peer review, independent reproduction or formal verification); numerical checks are floating-point sanity checks, not certificates or proofs. Creator: Anonymous. Text CC0-1.0; code MIT.
Author: Anonymous
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225276005