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The Influence Function of Transport-based Quantiles

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Pith's one-line read In every dimension d≥2, the influence function of a transport-based quantile is unbounded: it diverges like the inverse (d−1)-power of the distance from the quantile level, so infinitesimal point-mass contamination at an inlier moves the qu

desk verdict First rigorous characterization of transport-quantile influence functions: a pole-type singularity in d≥2, with a sound PDE proof but a few presentation gaps. read the letter →

arxiv 2607.19080 v1 pith:XGMIAV4C submitted 2026-07-21 math.ST stat.MEstat.TH

Alberto González-Sanz , Shunan Sheng , Bohan Wu , Marco Avella Medina This is my paper
classification math.STstat.MEstat.TH MSC 62G3562G30
keywords transport-basedquantilesinfluencefunctionrobuststatisticsoptimaltransportpolesingularityheavy-tailedasymptoticsempiricalfundamental-solutionpotential
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how much a transport-based (optimal-transport) multivariate quantile changes when the underlying distribution is contaminated by an infinitesimal point mass. It proves that in every dimension d≥2 the influence function exists away from the contamination point and is given by the gradient of the fundamental-solution potential of a uniformly elliptic operator. That potential has a pole-type singularity: at a fixed quantile level z, the influence of contaminating at x0 grows like ||z − F_P(x0)||^{-(d−1)}, where F_P is the transport distribution function. Consequently the influence function is not square-integrable under P, so the usual √n Gaussian asymptotic linear representation cannot hold with a square-integrable influence function; the paper conjectures slower n^{1/d} (for d≥3) and √(n/log n) (for d=2) rates and gives numerical evidence of stable-type, non-Gaussian fluctuations. If correct, this marks a fundamental robustness difference between univariate quantiles, whose influence function is bounded, and multivariate transport quantiles.

What carries the argument

The carrying object is the potential G_{x0}, the fundamental-solution potential of the uniformly elliptic operator div(f_μ[∇Q_P]^{-1}∇·) with a Dirac source and a no-flux boundary condition; the influence function is exactly ∇G_{x0}. Uniform ellipticity follows from classical regularity theory for optimal maps between regular measures, and it is what lets the proof freeze coefficients near the singularity, compare G_{x0} with the explicit fundamental solution of a constant-coefficient operator, and extract the ||z − F_P(x0)||^{-(d−1)} rate. A second mechanism is the shape control of the preimage K_{x0,t} of the contamination point under the contaminated transport map: an ellipsoid lemma show

What would settle it

In the explicit uniform-ball example μ=P=U(B_1(0)), compute or simulate the influence function I(x;0) stated in Proposition 4.1 for x approaching 0: it must scale exactly as ||x||^{-(d−1)} and its L^2(P) norm must diverge. Alternatively, estimate the tail of the empirical linear statistic in Conjecture 4.4 at increasing sample sizes; if the correct normalizer deviates from n^{1/d} (or √(n/log n) in d=2) or the limiting distribution is Gaussian, the pole-type singularity claim is contradicted.

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Extended reading notes

Core claim

For regular probability measures μ and P, the influence function I(x0; Q_P(z)) exists for every x0 ≠ Q_P(z) and equals ∇G_{x0}(z), where G_{x0} is the unique zero-mean solution of div(f_μ[∇Q_P]^{-1}∇G) = μ − δ_{F_P(x0)} with a no-flux (homogeneous Neumann) boundary condition on the reference domain. Near the singularity the influence function obeys the two-sided estimate ||I(x0; Q_P(z))|| ≍ ||z − F_P(x0)||^{-(d−1)}. In words, the transport quantile is infinitesimally sensitive to perturbations at points whose transport-distribution coordinate is close to the queried level; the sensitivity is a pole, not a jump. The proof combines L^2 stability of the contaminated optimal transport map, unifo

Load-bearing premise

Everything rests on the regularity assumption that both the reference and target measures have bounded convex C^{2,1} support with C^{1,α} densities bounded away from zero and infinity; if densities vanish at the boundary or lack this smoothness, the uniform ellipticity that produces the exact pole rate can break down.

Editorial extensions

If this is right

  • In dimensions d≥2, transport quantiles are not robust in the classical bounded-influence sense: contamination at an inlier close to the quantile level produces unbounded first-order sensitivity, even though transport quantiles have high breakdown points.
  • The influence function lies in L^q(P) for every q < d/(d−1) but not in L^2(P), so a standard √n asymptotically linear representation with a square-integrable influence function is impossible.
  • The paper's conjecture gives the scaling n^{1/d} for d≥3 and √(n/log n) for d=2 for empirical transport quantiles, with numerical experiments showing the empirical quantile and the influence-function linear statistic aligning under this scaling.
  • For d≥3, the preimage of the contamination point under the contaminated map is asymptotically a ball of radius t^{1/d}; in dimension two it is contained in a ball of radius √(t|log t|).
  • The symmetry G_x(z) = G_{Q_P(z)}(F_P(x)) provides a practical finite-difference recipe for computing the influence function at any quantile level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step is to test whether trimming or capping observations near the quantile level restores √n Gaussian limits; the pole rate suggests the trimming radius should scale with n^{-1/d} in dimension d.
  • The same elliptic fundamental-solution mechanism likely governs other point-mass-perturbed optimal-transport functionals, so pole-type influence with exponent d−1 may be a general phenomenon for transport-based statistics, not just for quantiles.
  • If the conjecture's stable-type limit is correct, bootstrap confidence intervals built on normal approximations will undercover for empirical transport quantiles; simulations should show coverage degrading as n grows unless intervals are based on stable quantiles.
  • The tail index γ = d/(d−1) is directly testable: estimate the tail of the empirical linear statistic n^{1/d}∑I(X_i; Q_P(z)) at increasing n and compare with a γ-stable fit; Proposition 4.5 already provides a rigorous testing-functional version that may extend to the full convergence.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies the influence function (IF) of transport-based quantiles, i.e., of the optimal transport map Q_P pushing a fixed reference measure mu to a target P. For Huber contaminations P_t=(1-t)P+t delta_{x_0}, it proves that the limit I(x_0;Q_P(z)) = lim_{t->0}(Q_{P_t}(z)-Q_P(z))/t exists for x_0 != Q_P(z), that it admits the PDE representation I(x_0;Q_P(z)) = nabla G_{x_0}(z), where G_{x_0} solves a uniformly elliptic Neumann problem with a Dirac source, and that in dimension d>=2 this IF has a pole-type singularity: ||I(x_0;Q_P(z))|| is comparable to ||z-F_P(x_0)||^{-(d-1)} locally. The paper also shows the IF has infinite second moment under P, presents an explicit uniform-ball example, and gives numerical evidence for a conjectured n^{1/d} (d>=3) or sqrt(n/log n) (d=2) scaling of empirical transport quantiles.

Significance. If established, this is an important and somewhat surprising result: it shows that transport quantiles in dimension d>=2 are infinitesimatically sensitive to inliers, in contrast to bounded univariate quantile IFs, and it provides a rigorous PDE route to influence analysis for nonsmooth perturbations. The proof strategy is substantial: it combines an L^2 stability estimate, upgraded L^q bounds, local maximum principle/Moser iteration, Schauder estimates, and a comparison with the frozen fundamental solution. The explicit uniform-ball computation and the symmetry lemma for the Green potential are valuable concrete contributions. The falsifiable predictions -- the exact singularity rate, infinite second moment, and the conjectured empirical scaling -- are clearly stated and numerically probed. The paper is careful to distinguish the unbounded IF from high breakdown point, which is a common source of confusion.

major comments (2)
  1. [Section 5.1, Eq. (16)] Theorem 5.1 is the cornerstone of the proof, but the key inequality (16) is quoted from (Manole et al., 2024, Theorem 6) and then applied to P_t=(1-t)P+t delta_{x_0}, which is not absolutely continuous. The manuscript does not state the hypotheses of that theorem. If the theorem requires regular or absolutely continuous targets, the application to an atomic perturbation is not justified. Please either restate the precise theorem and verify its applicability to atomic P_t (e.g., by approximation), or give a self-contained proof of the L^2 stability bound for atomic targets.
  2. [Section 5.6, Proposition 3.5] The d=2 case of Proposition 3.5 is explicitly omitted ('The case d=2 follows from a similar argument and the details are omitted'). This is not merely cosmetic: Proposition 3.5 is used in Lemma 5.6 to ensure K_{t,x0} is disjoint from a fixed open set U, which is then used in Proposition 5.5 and hence in the proof of Theorem 3.1 for d=2. Since the paper's headline claim covers all d>=2, the d=2 proof should be written out or a complete reference should be provided.
minor comments (3)
  1. [Section 4.1 / Proof of Proposition 4.1(iii)] In the proof of Proposition 4.1(iii), the influence function is written as I(x;z)=nabla^2 G_x(z); the Hessian should be a gradient, I(x;z)=nabla G_x(z). The displayed formula and the surrounding text should be corrected.
  2. [Section 5.5, Step 2.2] The estimate for the Hoelder seminorm [f_0]_{C^alpha(B_{R/4}(z*))} is asserted by direct computation but not shown. Given the importance of the scaling in Eq. (60), a few lines of derivation would improve checkability.
  3. [General] Several acronyms and names have inconsistent accents/diacritics (e.g., 'Monge-Ampere' vs 'Monge-Ampère'). The paper would also benefit from a statement near Eq. (16) that the implicit constants in the L^2 estimates are uniform in the small-t regime.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the influence function is derived from a PDE analysis, not from fitting or self-citation.

full rationale

The paper's central object is the influence function I(x0;Q_P(z)) = lim_{t↓0} [Q_{P_t}(z)−Q_P(z)]/t, characterized in Theorem 3.1 as ∇G_x0 where G_x0 solves div(fμ[∇Q_P]^{-1}∇G)=μ−δ_{z0}. This characterization is proved via L^2 and L^q estimates (Theorem 5.1, Proposition 5.3), local C^{2,α} bounds (Proposition 5.5), compactness, and a uniqueness argument (Proposition 5.7). The pole-type singularity of Theorem 3.3 follows by comparing G_x0 with the constant-coefficient fundamental solution Φ_x0 and controlling the remainder v via the local maximum principle, Schauder estimates, and the uniform L^p bound in Lemma 5.8; the rate ∥z−F_P(x0)∥^{−(d−1)} is derived, not assumed. The symmetry identity G_x(z)=G_{Q_P(z)}(F_P(x)) is proved in Lemma 5.10 from the PDE rather than imported. The explicit uniform example uses Wirth's external Green-function formula, and the L^2 stability input (16) is an external theorem from Manole et al. (2024), whose authors do not overlap with this paper. Self-citations (e.g., Avella Medina and González-Sanz 2026; Gonzalez-Sanz and Avella Medina 2026; González-Sanz and Sheng 2024) concern breakdown points or are technique references; they are not the load-bearing derivation of the IF singularity. Conjecture 4.4 is explicitly labeled a conjecture and is not presented as a theorem. Thus there is no equation or parameter equivalent to its inputs by construction, no fitted prediction renamed as a result, and no self-citation chain forcing the central claim.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation depends on regularity assumptions and external PDE/OT results, but introduces no parameters fitted to data and no new physical or mathematical entities. The Green-type potential G_{x0} is a solution of an existing PDE, not an ad hoc construction.

assumptions (5)
  • standard math Brenier's theorem (Theorem 2.1): existence/uniqueness of the OT map as the gradient of a convex function.
    Invoked in Section 2.2 to define Q_P and F_P and to use Caffarelli regularity.
  • standard math Caffarelli's global regularity (Theorem 2.3): for regular μ,P, Q_P,F_P ∈ C^{2,α} with uniform ellipticity of ∇Q_P.
    Used pervasively in Sections 3 and 5 to set up the uniformly elliptic PDE and to localize the fundamental solution.
  • standard math L2–W2 stability bound (Manole et al. 2024, Theorem 6): ||Q_{P_t}−Q_P||²_{L²(μ)} ≲ W₂²(P_t,P).
    External theorem used in the proof of Theorem 5.1, the cornerstone of the L2 estimate.
  • standard math Elliptic regularity, Schauder estimates, local maximum principle, and John's lemma (Gilbarg–Trudinger; Taira; Figalli; Gutiérrez).
    Foundational PDE results used in Section 5 and Appendix A to upgrade convergence and bound the Green function.
  • domain assumption Definition 2.2 regularity assumption: μ and P have bounded convex C^{2,1} support and C^{1,α} densities bounded away from 0 and ∞.
    This is the structural hypothesis on the measures; without it the ellipticity and regularity arguments in Theorems 3.1 and 3.3 break down.

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Pith. "Pith review of The Influence Function of Transport-based Quantiles." pith.science (2026). https://pith.science/paper/XGMIAV4C

@misc{pith2026260719080,
  author       = {Pith},
  title        = {Pith review of: The Influence Function of Transport-based Quantiles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XGMIAV4C}},
  note         = {Machine review of arXiv:2607.19080}
}
abstract

Transport-based quantiles extend univariate quantiles to multivariate distributions via optimal transport. We study the influence function of the transport quantile map $\mathbf{Q}_P$, defined as the optimal transport map pushing a fixed reference measure $\mu$ forward to a target distribution $P$. For the Huber contamination $P_t=(1-t)P+t\delta_{x_0}$, we prove that the first-order limit $\mathbf{I}(x_0;\mathbf{Q}_P(z)) := \lim_{t\downarrow 0} [\mathbf{Q}_{(1-t)P+t\delta_{x_0}}(z)-\mathbf{Q}_P(z)]/t$ exists whenever $x_0\ne \mathbf{Q}_P(z)$ and characterize it uniquely. Specifically, $\mathbf{I}(x_0;\mathbf{Q}_P(z))=\nabla G_{x_0}(z)$, where $G_{x_0}$ is characterized by a uniformly elliptic equation with a Dirac source and a Neumann boundary condition. In every dimension $d\ge 2$, this influence function has a pole-type singularity. For fixed $z\in\operatorname{int}(\Omega_\mu)$, it remains bounded when $\mathbf{F}_P(x_0)$ stays away from $z$, where $\mathbf{F}_P=\mathbf{Q}_P^{-1}$ is the transport-based distribution function, but diverges as $x_0\to\mathbf{Q}_P(z)$, equivalently as $\mathbf{F}_P(x_0)\to z$. In fact, $\|\mathbf{I}(x_0;\mathbf{Q}_P(z))\|\asymp\|z-\mathbf{F}_P(x_0)\|^{-(d-1)}$. This contrasts with the bounded influence function of univariate quantiles and implies that $\mathbf{I}(X;\mathbf{Q}_P(z))$, for $X\sim P$, has infinite second moment. Numerical experiments further suggest that empirical transport quantiles may exhibit stable-type non-Gaussian fluctuations.

Figures

Figures reproduced from arXiv: 2607.19080 by the authors.

Figure 1
Figure 1. Flow Chart for the Proof of Main Results. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The left panel shows the norm of the influence function of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Empirical distributions of n 1/dQPn (0) (blue) and the linear statistics n −1+1/d P i I(Xi ; QP (0)) (orange), for n ∈ {250, 1000, 4000, 16000} (rows 1–4), d = 3, and R = 1000 replications. The two distributions remain stable as n grows and provide finite￾sample evidence consistent with the scaling in Conjecture 4.4. While the OT median looks normal for small sample sizes, as n increases the distribution of the empi… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Intuition behind Proposition 3.5. Inside the set Kt,x0 (green) we have the ellipsoid E (purple) and outside its dilation d 3/2E (orange). The goal in Proposition 3.5 is to show that the axes λ1,t ≥ λ2,t of E decrease with rate t 1 d for d ≥ 3. The following is the cele…

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  1. Empirical optimal transport potentials: fast rates and a functional central limit theorem

    math.ST 2026-08 accept novelty 8.0 of 10

    Empirical Brenier potentials converge in L1(μ) at rate n^{-1/2} for d≤3, n^{-1/2} log^{5/2} n for d=4, and n^{-2/d} log^{(d+2)/d} n for d≥5, with sharp polynomial exponents, an FCLT and consistent bootstrap for d≤3.

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