Horospherical Depth and Busemann Median on Hadamard Manifolds

2026-04-20 · Source: Takara TLDR - Daily AI Papers · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Researchers Yangdi Jiang, Xiaotian Chang, and Cyrus Mostajeran introduce horospherical depth, a new statistical depth concept for Hadamard manifolds, and define the Busemann median as its maximizers. This method leverages Busemann functions, which are limits of renormalized distance functions, to create horoballs that act as intrinsic half-spaces. The depth is parametrized by the visual boundary, is isometry-equivariant, and avoids tangent-space linearization or a base point. For any Hadamard manifold, the depth regions are nested and geodesically convex, ensuring a centerpoint of depth at least $1/(d+1)$ and the existence of a Busemann median for all Borel probability measures. Under strictly negative sectional curvature, the depth is strictly quasi-concave, leading to a unique median.

Key takeaway

For research scientists working with statistical analysis on non-Euclidean geometries, particularly Hadamard manifolds, adopting horospherical depth provides a robust and intrinsic measure of data centrality. This approach offers advantages over traditional methods by not requiring tangent-space linearization or a chosen base point, and its Busemann median is more stable against contamination than the Fréchet mean. You should consider integrating this depth definition into your geometric data analysis workflows for improved robustness and theoretical consistency.

Key insights

Horospherical depth offers an intrinsic, robust statistical depth for Hadamard manifolds, defining a unique Busemann median.

Principles

• Busemann functions replace linear functionals for depth on Hadamard manifolds.
• Depth regions are nested and geodesically convex on Hadamard manifolds.
• Strictly negative sectional curvature ensures unique Busemann medians.

Method

The method constructs horospherical depth using Busemann functions, whose sublevel sets form horoballs, serving as intrinsic half-spaces on Hadamard manifolds. This approach avoids tangent-space linearization and base point selection.

In practice

• Apply horospherical depth for robust statistical analysis on curved spaces.
• Utilize Busemann median for central tendency in non-Euclidean data.
• Leverage isometry-equivariance for consistent results across transformations.

Topics

• Horospherical Depth
• Busemann Median
• Hadamard Manifolds
• Busemann Functions
• Statistical Depth

Best for: AI Scientist, Research Scientist

Related on AIssential

• Barycentric Projections of Optimal Transport Plans on Riemannian Manifolds — A framework enables deterministic map extraction from optimal transport couplings on Riemannian manifolds using intrinsic and tangential projections.
• Jensen's Inequality - Why the Average of a Curve is Not the Curve of the Average — Convexity and Jensen's inequality reveal how function curvature dictates the relationship between function of expectation and expectation of function.
• On the Geometry of Separation in Finite Gaussian Mixtures — A geometric framework links Gaussian mixture separation and weight to parameter estimation convergence via Wasserstein distances, revealing configuration-dependent complexity.
• The Data Don’t Fill the Space — Manifold geometry analyzes data concentrated on a low-dimensional, curved surface embedded in a higher-dimensional space.
• The most beautiful formula not enough people understand — High-dimensional geometry reveals counterintuitive properties, with n-dimensional sphere volumes peaking then becoming infinitesimal.
• A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification — PLACE offers a closed-form pipeline for point cloud and graph classification with certified guarantees from persistent homology.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Takara TLDR - Daily AI Papers.