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arxiv: 2411.10646 · v3 · submitted 2024-11-16 · 🧮 math.ST · stat.ME· stat.TH

Wasserstein Spatial Depth

Fran\c{c}ois Bachoc , Alberto Gonz\'alez-Sanz , Jean-Michel Loubes , Yisha Yao This is my paper

Pith reviewed 2026-05-23 17:31 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords statistical depthWasserstein distancedistribution-valued dataspatial depthgeometric medianrobustnessclusteringtwo-sample test
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The pith

Wasserstein spatial depth extends classical statistical depth to probability distributions equipped with the Wasserstein metric.

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

The paper introduces Wasserstein spatial depth to rank and order entire probability distributions rather than scalar or vector observations. It adapts the spatial depth construction to the geometry of the Wasserstein space on distributions over R^d for d greater than 1. A reader would care because the resulting measure supports order-based methods such as clustering and two-sample testing directly on distribution-valued data. The construction is shown to keep the standard depth axioms intact, including values in [0,1], geodesic invariance, vanishing at infinity, and maximization at the geometric median.

Core claim

We extend the concept of statistical depth to distribution-valued data, introducing the notion of Wasserstein spatial depth. This new measure provides a way to rank and order distributions, enabling the development of order-based clustering techniques and inferential tools. We show that Wasserstein spatial depth (WSD) preserves critical properties of conventional statistical depths, notably, ranging within [0,1], transformation and geodesic invariance, vanishing at infinity, reaching a maximum at the geometric median, and continuity. The population WSD admits a plug-in estimator based on empirical distributions that is consistent and asymptotically normal, and the approach yields a twoSample

What carries the argument

Wasserstein spatial depth, obtained by replacing Euclidean distance in the classical spatial depth formula with the geodesic distance induced by the Wasserstein metric on the space of distributions.

If this is right

  • The depth supplies a ranking that directly yields order-based clustering algorithms for populations of distributions.
  • A two-sample test for equality of distribution populations follows from comparing depth values or depth regions.
  • The empirical depth regions have explicitly characterized breakdown points under contamination.
  • The plug-in estimator based on sampled empirical distributions is consistent and asymptotically normal.
  • The influence function of the depth can be derived to quantify robustness to perturbations of individual distributions.

Where Pith is reading between the lines

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

  • The same construction could be attempted with other optimal-transport metrics provided they induce a geodesic structure compatible with the depth axioms.
  • Applications to functional data or to point processes become immediate once those objects are embedded in a Wasserstein-type space.
  • One could test whether depth-based ordering improves upon moment-based or kernel-based methods on concrete tasks such as image histogram classification.

Load-bearing premise

The space of distributions under the Wasserstein metric behaves enough like a Riemannian manifold that the Euclidean spatial depth formula extends directly while keeping all listed invariance and extremal properties.

What would settle it

A finite collection of distributions whose empirical Wasserstein geometric median does not maximize the computed Wasserstein spatial depth value, or a sequence of distributions converging in Wasserstein distance whose depth values fail to converge.

Figures

Figures reproduced from arXiv: 2411.10646 by Alberto Gonz\'alez-Sanz, Fran\c{c}ois Bachoc, Jean-Michel Loubes, Yisha Yao.

Figure 1
Figure 1. Figure 1: The green solid lines depict the change of theoretical WSD along the [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The relationships between the WSD and conventional spatial depth in [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left panel: the distributions are drawn according to Case 1. Right panel: [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a): The data points are drawn from the distributions of Case 1. The [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a): The data points are drawn from the distributions of Case 2. The [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The green dots represent regular/representative/central distributions, [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparisons between the most regular years 1935 and 1960 (the two [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
read the original abstract

Modeling observations as random distributions embedded within Wasserstein spaces is becoming increasingly popular across scientific fields, as it captures the variability and geometric structure of the data more effectively. However, the distinct geometry and unique properties of Wasserstein spaces pose challenges to the application of conventional statistical tools, which are primarily designed for Euclidean spaces. Consequently, adapting and developing new methodologies for analysis within Wasserstein spaces has become essential. The space of distributions on $\mathbb{R}^d$ with $d>1$ is not linear, and "mimic" the geometry of a Riemannian manifold. In this paper, we extend the concept of statistical depth to distribution-valued data, introducing the notion of Wasserstein spatial depth. This new measure provides a way to rank and order distributions, enabling the development of order-based clustering techniques and inferential tools. We show that Wasserstein spatial depth (WSD) preserves critical properties of conventional statistical depths, notably, ranging within $[0,1]$, transformation and geodesic invariance, vanishing at infinity, reaching a maximum at the geometric median, and continuity. Regarding robustness, we characterize the breakdown points of the empirical depth regions and the influence function of the WSD. Additionally, the population WSD has a straightforward plug-in estimator based on sampled empirical distributions. We establish the estimator's consistency and asymptotic normality. We also provide a two-sample test for populations of distributions based on the WSD. Finally, extensive simulations and a real-data application showcase the practical efficacy of the WSD.

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

3 major / 1 minor

Summary. The paper introduces Wasserstein spatial depth (WSD) for distribution-valued data in the Wasserstein space of probability measures on R^d (d>1). It defines WSD by direct analogy to Euclidean spatial depth, leveraging the claim that the Wasserstein metric space 'mimics the geometry of a Riemannian manifold.' The manuscript asserts that WSD satisfies the standard depth axioms (range in [0,1], transformation and geodesic invariance, vanishing at infinity, unique maximum at the geometric median, continuity), characterizes breakdown points of empirical depth regions and the influence function, establishes consistency and asymptotic normality of the plug-in estimator, and proposes a two-sample test, supported by simulations and a real-data example.

Significance. If the geometric construction is made explicit and the claimed invariance and extremal properties are shown to hold without additional assumptions that reduce the result to a fitted quantity, the work would supply a concrete tool for ranking and clustering distributions with robustness and inferential guarantees. The plug-in estimator's asymptotic normality and the two-sample test would be the most immediately usable contributions for applications involving distribution-valued observations.

major comments (3)
  1. [Abstract] Abstract and opening paragraphs: the definition of WSD is justified by the statement that the Wasserstein space 'mimics the geometry of a Riemannian manifold,' yet no explicit construction (tangent-space projection, exponential map, or handling of non-unique geodesics and cut loci) is supplied. Without this, the transfer of the listed properties (geodesic invariance, unique maximum at the geometric median) does not follow automatically for d>1.
  2. [Abstract] Abstract: the claims of preservation of depth axioms, breakdown-point calculations, influence function, consistency, and asymptotic normality are asserted without any displayed derivations, explicit formulas, or theorem statements. This prevents verification that the properties hold under the actual Wasserstein geometry rather than by construction.
  3. [Introduction / Definition] The weakest modeling assumption (that the Wasserstein metric space admits a direct, well-behaved extension of Euclidean spatial depth) is load-bearing for every subsequent claim; if the construction fails to be well-defined everywhere, the robustness, consistency, and test results rest on an unverified foundation.
minor comments (1)
  1. [Abstract] Notation for the Wasserstein metric and the geometric median should be introduced with a displayed equation at first use to avoid ambiguity when d>1.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive report and the opportunity to clarify the geometric foundation and presentation of our results on Wasserstein spatial depth. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the definition of WSD is justified by the statement that the Wasserstein space 'mimics the geometry of a Riemannian manifold,' yet no explicit construction (tangent-space projection, exponential map, or handling of non-unique geodesics and cut loci) is supplied. Without this, the transfer of the listed properties (geodesic invariance, unique maximum at the geometric median) does not follow automatically for d>1.

    Authors: We agree that the phrasing 'mimics the geometry of a Riemannian manifold' is informal and that an explicit metric-space definition is preferable. The WSD is defined directly via the Wasserstein distance in the spatial-depth formula (Section 2), replacing the Euclidean norm with W_2 without invoking tangent spaces or exponential maps. Geodesic invariance follows from the isometry property of the Wasserstein metric under push-forwards, and the unique maximum at the geometric median is proved using the strict convexity of the Wasserstein distance for measures with finite second moments. For d>1 we will add a remark noting that non-uniqueness of geodesics does not affect the distance-based definition. These clarifications and the explicit formula will be inserted in the revised introduction. revision: yes

  2. Referee: [Abstract] Abstract: the claims of preservation of depth axioms, breakdown-point calculations, influence function, consistency, and asymptotic normality are asserted without any displayed derivations, explicit formulas, or theorem statements. This prevents verification that the properties hold under the actual Wasserstein geometry rather than by construction.

    Authors: The abstract is a high-level summary; the full statements appear as Theorems 3.1 (depth axioms), 4.1–4.2 (breakdown points and influence function), 5.1 (consistency), and 5.2 (asymptotic normality) with proofs in the appendix. To improve readability we will move the theorem statements (without proofs) into a new subsection of the introduction so that the abstract claims are immediately traceable to the stated results. revision: yes

  3. Referee: [Introduction / Definition] The weakest modeling assumption (that the Wasserstein metric space admits a direct, well-behaved extension of Euclidean spatial depth) is load-bearing for every subsequent claim; if the construction fails to be well-defined everywhere, the robustness, consistency, and test results rest on an unverified foundation.

    Authors: The definition in Section 2 is given for all probability measures with finite second moments (the natural domain of W_2), and well-definedness follows immediately from the triangle inequality and continuity of W_2. All subsequent results (robustness, consistency, two-sample test) are proved from this metric definition using only properties of the Wasserstein space as a complete separable metric space; no additional Riemannian structure is assumed. We will add a short paragraph after the definition explicitly stating the domain and confirming that the extension is well-defined everywhere on this domain. revision: partial

Circularity Check

0 steps flagged

No circularity: definition of WSD followed by independent proofs of listed properties

full rationale

The paper defines Wasserstein spatial depth by direct analogy to the Euclidean spatial depth using the Wasserstein metric on the space of distributions. It then states that it shows the measure preserves the standard properties (range [0,1], invariances, vanishing at infinity, maximum at geometric median, continuity). No quoted step reduces any claimed property to a fitted input, self-definition, or self-citation chain; the listed properties are presented as results to be established rather than presupposed in the definition. The modeling choice that the Wasserstein space 'mimics' Riemannian geometry is an explicit modeling assumption used to motivate the definition, not a hidden reduction. This is the normal case of a self-contained construction with subsequent verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that Wasserstein geometry supports a depth function with the classical Euclidean properties; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The space of probability distributions equipped with the Wasserstein metric admits a depth notion that inherits the standard properties of statistical depth (range [0,1], invariance, maximum at geometric median).
    Invoked when the authors state that the space 'mimics the geometry of a Riemannian manifold' and proceed to define and prove properties of WSD.

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