Horospherical Depth and Busemann Median on Hadamard Manifolds

2026-04-20

AI Summary

This work addresses the absence of an intrinsic, isometry-equivariant statistical depth on Hadamard manifolds that avoids linearization in tangent spaces. The authors propose a novel depth based on Busemann functions, replacing the halfspaces in Tukey depth with horoballs to construct an intrinsic depth independent of basepoint selection, and define its maximizer as the Busemann median. This approach is the first to leverage the visual boundary for intrinsic depth construction, ensuring isometry equivariance and, under negative curvature, guaranteeing uniqueness of the median and robustness against contamination at infinity. Theoretical analysis shows that depth regions are nested and geodesically convex, and that a central point with depth at least $1/(d+1)$ always exists. Furthermore, the sample depth converges uniformly, and the sample median converges both in symmetric spaces and general Hadamard manifolds.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

\We introduce the horospherical depth, an intrinsic notion of statistical depth on Hadamard manifolds, and define the Busemann median as the set of its maximizers. The construction exploits the fact that the linear functionals appearing in Tukey's half-space depth are themselves limits of renormalized distance functions; on a Hadamard manifold the same limiting procedure produces Busemann functions, whose sublevel sets are horoballs, the intrinsic replacements for halfspaces. The resulting depth is parametrized by the visual boundary, is isometry-equivariant, and requires neither tangent-space linearization nor a chosen base point.For arbitrary Hadamard manifolds, we prove that the depth regions are nested and geodesically convex, that a centerpoint of depth at least $1/(d+1)$ exists, and hence that the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique. We also establish robustness: the depth is stable under total-variation perturbations, and under contamination escaping to infinity the limiting median depends on the escape direction but not on how far the contaminating mass has moved along the geodesic ray, in contrast with the Fréchet mean. Finally, we establish uniform consistency of the sample depth and convergence of sample depth regions and sample Busemann medians; on symmetric spaces of noncompact type, the argument proceeds through a VC analysis of upper horospherical halfspaces, while on general Hadamard manifolds it follows from a compactness argument under a mild non-atomicity assumption.

Problem

• statistical depth
• Hadamard manifolds
• Busemann median
• non-Euclidean statistics
• robust estimation

Innovation

• horospherical depth
• Busemann median
• Hadamard manifolds
• statistical depth
• robustness