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The scalar spatial depth alone completely determines any probability measure on a Hilbert space.

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2026-07-11 19:37 UTC pith:VQ4ZZ3MM

load-bearing objection They close two long-open characterization problems for spatial depth/quantiles, including the surprising scalar-depth result even in finite dimension, with clean reusable proofs.

arxiv 2607.04375 v1 pith:VQ4ZZ3MM submitted 2026-07-05 math.ST math.PRstat.TH

Spatial depth characterizes probability measures

Alberto Gonz\'alez-Sanz , Dimitri Konen This is my paper
classification math.ST math.PRstat.TH MSC 62G0562H0546N3049J53
keywords spatial depthgeometric quantilescharacterization of measuresHilbert spacegradient flowconvex potentialnonparametric inference
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Spatial (or geometric) depth measures how central a point is relative to a probability distribution by looking at the length of a certain average direction vector. For decades it was open whether two different distributions could produce exactly the same depth values everywhere. This paper proves they cannot: if the spatial depths of two Borel probability measures on a separable Hilbert space coincide at every point, the measures are identical. The same uniqueness holds for the full vector-valued spatial distribution function, even when equality is required only on a dense set and even in infinite dimension. The result supplies the missing theoretical guarantee that nonparametric procedures built from spatial depth or spatial quantiles are in principle able to recover the whole law, and the proof technique extends to other depths that arise as differentials of convex functionals.

Core claim

If two Borel probability measures P and Q on a separable Hilbert space satisfy ||F^g_P(x)|| = ||F^g_Q(x)|| for every x, then P equals Q. The same conclusion holds when the vector-valued maps F^g_P and F^g_Q themselves coincide, even if equality is only required on a dense subset. In infinite dimension this settles the open characterization problem for the spatial distribution function and its quantiles; the depth characterization was previously open even in finite dimension.

What carries the argument

The convex potential h_P(x) = ∫ (‖x-z‖-‖z‖) dP(z) whose subgradient contains the spatial distribution function F^g_P; comparison of h_P-h_Q along the unique gradient flow of Φ = h_P + h_Q shows that equal depths force the potentials (and therefore the measures) to coincide.

Load-bearing premise

The existence and uniqueness of the gradient flow for the continuous convex coercive potential formed by the sum of the two spatial potentials, for completely arbitrary Borel probability measures.

What would settle it

Exhibit two distinct Borel probability measures on a Hilbert space (even R^2) whose spatial depth functions coincide at every point, or show that the gradient flow of their joint potential fails to exist or to be unique.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Any statistical procedure that recovers the spatial depth function (or the spatial quantile map) recovers the entire underlying law.
  • Depth-based rank tests, classification rules and outlier diagnostics based on spatial depth are theoretically complete: distinct laws cannot produce identical depth fields.
  • The same uniqueness argument applies to the transport-based depth of Hallin et al., giving a second depth that fully characterizes measures.
  • Under a mild finite-dimensional marginal density condition the spatial quantile map is a C^∞ diffeomorphism, so quantile contours are smooth manifolds even in infinite dimension.

Where Pith is reading between the lines

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

  • The same convex-flow technique is likely to settle characterization questions for other depths that can be written as minimal-norm elements of subdifferentials of convex potentials.
  • Because the argument never uses finite dimensionality except for a few measure-theoretic technicalities, analogous uniqueness statements should hold on a large class of Banach spaces once a suitable notion of spatial cdf is fixed.
  • Empirical spatial depth estimators that converge uniformly will automatically be consistent for the whole law, opening a direct route to goodness-of-fit tests based solely on depth.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper establishes two characterization results for spatial (geometric) objects on separable Hilbert spaces H (finite- or infinite-dimensional). Theorem 1.1 shows that the spatial distribution function F^g_P (and hence the associated spatial quantiles when invertible) uniquely determines any Borel probability measure P, extending Koltchinskii’s finite-dimensional result via an asymptotic coordinate-perturbation argument, Carlson’s theorem, and Fourier uniqueness of measures on the line. Theorem 1.2 shows, more surprisingly, that the scalar spatial depth SD(x;P)=1-∥F^g_P(x)∥ also characterizes P: equality of depths everywhere forces P=Q, while equality on a dense set permits only a fully classified family of two-point (or, in dimension 1, median-swapped) exceptions. Theorem 1.3 further proves that a single finite-dimensional marginal with locally bounded density already makes F^g_P a C^∞-diffeomorphism onto the open unit ball. The proofs rely on the convex potential h_P, gradient-flow comparison of h_P-h_Q, and careful handling of atoms.

Significance. If correct, the results close long-standing open questions (infinite-dimensional characterization of F^g_P since Koltchinskii 1997; depth characterization even in finite dimension since Chaudhuri 1996 and Vardi–Zhang 2000) and supply the missing theoretical foundation for nonparametric depth-based inference, ranks, and quantiles in Hilbert spaces. The techniques—coordinate perturbation at infinity combined with Carlson’s theorem, and gradient-flow comparison of convex potentials adapted from Pérez-Aros–Salas–Vilches—are novel and immediately applicable to other depths (e.g., the transport-based depth of Hallin et al.). Full, self-contained proofs are given for all three main theorems, including the exceptional atomic cases, which strengthens the contribution.

minor comments (5)
  1. Page 2, display (3): the subtracted-norm construction of h_P is standard but could be briefly cross-referenced to Kemperman or Konen–Paindaveine for readers unfamiliar with the moment-free definition.
  2. Lemma 2.2: the appeal to Zorn’s lemma is correct, yet a short remark that a constructive orthonormal basis avoiding the countable set A_{P,Q} can be built by successive Gram–Schmidt on a dense countable set would improve accessibility.
  3. Section 3, after (10): the explicit description of the subdifferential of h_P is useful; adding a one-line citation to the corresponding formula in Konen–Stupfler (2026) would help the reader locate the earlier derivation.
  4. Theorem 1.3 / Corollary 4.4: the condition that a single finite-dimensional marginal has a locally L^p density is surprisingly weak; a short example (e.g., a Gaussian measure with non-degenerate covariance) illustrating that the resulting quantile contours are C^∞ would make the result more vivid.
  5. References: a few arXiv preprints (Romon 2022, Passeggeri–Reid 2022, Konen 2025a) are listed without final publication data; updating them if available would be desirable.

Circularity Check

1 steps flagged

Minor self-citations to authors' prior technical lemmas on spatial quantiles; central infinite-dimensional and depth characterizations are derived from first principles via novel Fourier/Carlson and gradient-flow arguments.

specific steps
  1. self citation load bearing [Section 2, finite-dimensional case of Theorem 1.1]
    "In particular, Theorem 3.2 in Konen (2025a) entails that P=Q on the Borel subsets of R^d. This yields the conclusion of Theorem 1.1 when H is finite-dimensional."

    The finite-dimensional uniqueness of the spatial cdf is imported from the first author's own prior work rather than re-derived or taken from the classical Koltchinskii (1997) reference already cited in the introduction. The step is not load-bearing for the paper's main novelty (infinite-dimensional case and depth characterization), so the circularity is minor.

full rationale

The paper's two main theorems (spatial cdf characterizes P even in infinite-dimensional Hilbert spaces; the scalar spatial depth ||F^g_P|| fully characterizes P) are proved from the definitions of F^g_P as the Bochner integral of the unit vector field and of h_P as the convex potential whose subgradient contains F^g_P. Theorem 1.1 proceeds by asymptotic coordinate perturbation, extension to a holomorphic function, Carlson's theorem, and Fourier uniqueness of the resulting radial measures; these steps are self-contained and do not reduce to any input assumption. Theorem 1.2 compares h_P - h_Q along the gradient flow of the continuous convex coercive potential Φ = h_P + h_Q (standard existence/uniqueness from Attouch–Buttazzo–Michaille), carefully handling atoms via a countable exceptional set S and classifying the two-point exceptional measures; the argument adapts (but does not merely invoke) the minimal-norm subgradient idea of Pérez-Aros et al. Self-citations appear only for intermediate technical facts (continuity of F^g_P at non-atoms, finite-dimensional distributional uniqueness, directional derivatives of h_P, invertibility of the Fréchet derivative of F^g_P). None of these citations is load-bearing for the novel claims, nor do they import an unverified uniqueness theorem that forces the result. There are no fitted parameters, no data-driven predictions, no smuggled ansatzes, and no renaming of known empirical patterns. The derivation chain is therefore essentially non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside standard real analysis, convex analysis and probability on separable Hilbert spaces. No free parameters are fitted; no new physical or statistical entities are postulated. The only non-standard ingredients are the classical spatial-depth definitions (already in the literature) and the invocation of gradient-flow existence for continuous convex coercive functionals (standard in the Attouch–Buttazzo–Michaille monograph).

axioms (4)
  • domain assumption H is a separable real Hilbert space; P, Q are Borel probability measures on H.
    Stated at the opening of Section 1; all results are relative to this setting.
  • standard math Existence and uniqueness of the absolutely continuous gradient flow of a continuous convex coercive functional (Proposition A.1).
    Cited from Attouch–Buttazzo–Michaille; used as a black box throughout Section 3.
  • standard math Carlson’s theorem on analytic functions vanishing on the non-negative integers.
    Lemma 2.3, taken from Titchmarsh / Hardy; applied in the complex-extension step of the infinite-dimensional uniqueness proof.
  • standard math Zorn’s lemma to produce an orthonormal basis avoiding the countable atom set A_{P,Q}.
    Lemma 2.2; classical choice principle, used only to guarantee a convenient basis.

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Cite this review

Pith. "Pith review of Spatial depth characterizes probability measures." pith.science (2026). https://pith.science/paper/VQ4ZZ3MM

@misc{pith2026260704375,
  author       = {Pith},
  title        = {Pith review of: Spatial depth characterizes probability measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQ4ZZ3MM}},
  note         = {Machine review of arXiv:2607.04375}
}
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read the original abstract

We solve two open problems about spatial (or geometric) quantiles and depth. First we show that in infinite dimension, the spatial distribution function and the associated spatial quantiles characterize the underlying distribution, which has been established in Koltchinski (1997) in finite dimension but remained unknown in infinite dimension. Second, and more surprisingly, we show that the spatial depth also fully characterizes probability measures, which has been an open problem even in finite dimension since the introduction of these concepts in Chaudhuri (1996) and Vardi & Zhang (2000). Our results provide theoretical foundations for nonparametric depth-based statistical inference and introduce novel proof techniques to investigate these questions for other depth and quantile concepts.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Influence Function of Transport-based Quantiles

    math.ST 2026-07 conditional novelty 8.0

    The influence function of multivariate transport quantiles has a pole-type singularity in dimension ≥2, so contamination near a quantile level yields unbounded first-order sensitivity.

  2. Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator

    math.ST 2026-07 accept novelty 6.0

    The empirical spatial distribution and the plug-in spatial depth are uniformly L1-consistent over all of R^d with a dimension-free rate of 1/sqrt(n).

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This paper was first reviewed by grok-4.5 on July 11, 2026.