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arxiv: 2604.18242 · v2 · submitted 2026-04-20 · 🧮 math.ST · cs.LG· stat.ML· stat.TH

Horospherical Depth and Busemann Median on Hadamard Manifolds

Yangdi Jiang , Xiaotian Chang , Cyrus Mostajeran This is my paper

Pith reviewed 2026-05-14 21:44 UTC · model grok-4.3

classification 🧮 math.ST cs.LGstat.MLstat.TH
keywords horospherical depthBusemann medianHadamard manifoldsstatistical depthhoroballscenterpoint theoremrobust statisticsgeometric statistics
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The pith

Horospherical depth on Hadamard manifolds produces a Busemann median that exists for every Borel probability measure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper defines an intrinsic statistical depth called horospherical depth on Hadamard manifolds. It uses Busemann functions, which are limits of renormalized distance functions, to define horoballs as replacements for half-spaces in Tukey's depth. The depth is parametrized by the visual boundary and is equivariant under isometries without requiring a base point or tangent space linearization. The authors prove that depth regions are nested and geodesically convex for any Hadamard manifold, guaranteeing a centerpoint with depth at least 1/(d+1). This implies the Busemann median, the set of depth maximizers, exists for every Borel probability measure.

Core claim

The horospherical depth is defined using Busemann functions parametrized by the visual boundary, replacing half-spaces with horoballs. For any Hadamard manifold, the depth regions are nested and geodesically convex, ensuring the existence of a centerpoint with depth at least 1/(d+1) and thus the Busemann median for every probability measure. The construction is isometry-equivariant and intrinsic.

What carries the argument

Busemann functions and their associated horoballs as intrinsic analogs of half-spaces in the definition of statistical depth.

If this is right

  • Depth regions are nested and geodesically convex.
  • A centerpoint of depth at least 1/(d+1) exists.
  • The Busemann median exists for every Borel probability measure.
  • Under strictly negative sectional curvature the depth is strictly quasi-concave and the median is unique.
  • The depth is robust to total-variation perturbations and sample depth converges uniformly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This may allow robust median estimation in hyperbolic spaces without choosing a base point.
  • It contrasts with Fréchet means by having limiting behavior depending on escape direction but not distance.
  • The VC analysis on symmetric spaces suggests efficient computation for certain manifolds.
  • Extensions could apply similar limiting procedures to other geometric objects in statistics.

Load-bearing premise

The manifold is Hadamard so that Busemann functions are well-defined and horoballs behave like half-spaces.

What would settle it

A Hadamard manifold and probability measure where the maximum horospherical depth is less than 1/(d+1) for all points.

Figures

Figures reproduced from arXiv: 2604.18242 by Cyrus Mostajeran, Xiaotian Chang, Yangdi Jiang.

Figure 1
Figure 1. Figure 1: Horospheres and the Busemann function in the Poincaré ball model of hyperbolic geometry. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Opposite boundary directions through the same point [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: α-depth regions for symmetric distributions on H2 in the Poincaré disc model where α increases from outermost to innermost. Panel (a) shows a rotationally symmetric distribution and Panel (b) a centrally symmetric distribution, both centered at (0.35, 0.2). The common center, nested contours, and inherited symmetries visualize Theorem 4.2, Proposition 4.6, and the depth-region representation of Theorem 3.2… view at source ↗
Figure 4
Figure 4. Figure 4: Numerical illustrations of sampled-direction depth regions. Panel (a) shows zoomed-in [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustrative sampled-direction depth regions in [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
read the original 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 Fr\'echet 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces horospherical depth on Hadamard manifolds, constructed via Busemann functions (limits of renormalized distance functions) whose sublevel sets are horoballs serving as intrinsic half-spaces. The Busemann median is the set of maximizers of this depth. The authors prove that depth regions are nested and geodesically convex, that a centerpoint of depth at least 1/(d+1) exists for every Borel probability measure (hence the median exists), that the depth is strictly quasi-concave and the median unique under strict negative curvature plus mild regularity, and that the depth is robust to total-variation perturbations and to contamination escaping to infinity. Sample consistency and convergence of depth regions and medians are established, via VC analysis on symmetric spaces of noncompact type and via compactness under non-atomicity on general Hadamard manifolds.

Significance. If the central claims hold, the work supplies an intrinsic, isometry-equivariant statistical depth and median for Hadamard manifolds that requires neither a base point nor tangent-space linearization. The centerpoint theorem, the robustness contrast with Fréchet means under escaping contamination, and the consistency proofs (especially the VC argument on symmetric spaces) are substantive contributions to geometric statistics. The explicit separation of assumptions between symmetric spaces and general Hadamard manifolds is a strength.

major comments (2)
  1. [Theorem stating centerpoint existence] The centerpoint theorem (existence of a point of depth at least 1/(d+1) for arbitrary Borel measures on any Hadamard manifold) is load-bearing for the median existence claim. The manuscript should explicitly indicate which properties of the visual boundary and Busemann functions are used to adapt the standard convex-geometry argument, and confirm that geodesic convexity plus nesting alone suffice without further curvature restrictions.
  2. [Consistency theorem for general Hadamard manifolds] In the consistency result for general Hadamard manifolds, the mild non-atomicity assumption is essential for the compactness argument; it should appear verbatim in the theorem statement rather than only in the proof sketch, so that the precise scope is clear.
minor comments (2)
  1. [Abstract and introduction] The abstract refers to 'mild regularity assumptions' for uniqueness; these should be stated precisely (e.g., as a condition on the measure or on the curvature) already in the introduction.
  2. [Notation and definitions] Notation for the visual boundary and the parametrization of depth by boundary points should be introduced once and used uniformly; a short table of symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address the two major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The centerpoint theorem (existence of a point of depth at least 1/(d+1) for arbitrary Borel measures on any Hadamard manifold) is load-bearing for the median existence claim. The manuscript should explicitly indicate which properties of the visual boundary and Busemann functions are used to adapt the standard convex-geometry argument, and confirm that geodesic convexity plus nesting alone suffice without further curvature restrictions.

    Authors: We agree that the centerpoint result is central. The proof adapts the classical argument by using that depth regions are closed, geodesically convex, and nested; these properties follow directly from the fact that Busemann functions are convex and 1-Lipschitz, with sublevel sets being horoballs (which are geodesically convex). The visual boundary parametrizes the family of such half-spaces without requiring a base point. No additional curvature assumptions beyond the Hadamard definition (complete, simply connected, non-positive sectional curvature) are used. We will insert a short clarifying paragraph in the proof of the centerpoint theorem (and a corresponding remark in the introduction) that explicitly lists these properties and confirms that geodesic convexity plus nesting suffice. revision: yes

  2. Referee: In the consistency result for general Hadamard manifolds, the mild non-atomicity assumption is essential for the compactness argument; it should appear verbatim in the theorem statement rather than only in the proof sketch, so that the precise scope is clear.

    Authors: We thank the referee for this observation. The non-atomicity condition is indeed required for the compactness argument that yields convergence of sample depth regions and medians on general Hadamard manifolds. We will move the assumption verbatim into the statement of the relevant consistency theorem so that the precise hypotheses are stated at the theorem level rather than only in the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation defines horospherical depth directly via Busemann functions on Hadamard manifolds and proves nesting, geodesic convexity, and centerpoint existence (at least 1/(d+1)) from the geometry of horoballs and the visual boundary. These steps rely on intrinsic manifold properties rather than fitted parameters, self-referential equations, or load-bearing self-citations that reduce the result to its inputs. The consistency arguments distinguish symmetric spaces (VC dimension) from general cases (compactness + non-atomicity) without circularity. The construction is self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the geometric properties of Hadamard manifolds and the limiting behavior of renormalized distance functions; no free parameters are fitted and no new entities beyond the defined depth function are postulated.

axioms (2)
  • domain assumption Hadamard manifold: complete, simply connected Riemannian manifold with non-positive sectional curvature
    Required for Busemann functions to be convex and for horoballs to serve as intrinsic half-spaces; invoked throughout the construction and all theorems.
  • standard math Existence of visual boundary and Busemann functions as limits of renormalized distances
    Standard fact in Hadamard geometry used to replace linear functionals.
invented entities (2)
  • horospherical depth no independent evidence
    purpose: Intrinsic statistical depth function parametrized by the visual boundary
    Newly defined quantity whose maximizers form the Busemann median
  • Busemann median no independent evidence
    purpose: Set of maximizers of horospherical depth
    Defined directly from the new depth function

pith-pipeline@v0.9.0 · 5600 in / 1584 out tokens · 43474 ms · 2026-05-14T21:44:32.970690+00:00 · methodology

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Reference graph

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