{"id":"a913f399-9f11-4049-952e-51e7eff6c13a","arxiv_id":"2607.19080","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"Transport-based quantiles extend univariate quantiles to multivariate data using optimal transport. This paper proves that for dimension two and higher, these quantiles are infinitesimally sensitive to contamination at inlier points, with an unbounded influence function and infinite second moment.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central PDE argument is internally consistent under the stated regularity assumptions.","rationale":"I read the proof as a chain of standard PDE arguments. The weakest step in terms of verifiability is the quoted stability inequality, because P_t is not absolutely continuous while the surrounding text says 'when P, μ are regular.' All later steps (Lq bound, local C^{2,α}, Green-function comparison) are carried out in detail and check out. The reader's 'weakest assumption' about regularity is really a scope condition rather than an internal inconsistency: the paper explicitly restricts to regular measures and uses Caffarelli. The minor issues the reader noted (omitted d=2 argument, missing code/data, a typo) do not threaten the central theorem. Hence no verdict change.","tokens_in":43327,"tokens_out":37331,"duration_ms":371621,"concrete_test":"Verify the hypotheses of (Manole et al., 2024, Theorem 6) for the singular target Q=P_t=(1-t)P+tδ_{x0}. If the theorem is not stated to cover atomic Q, derive the L2 stability inequality directly for P_t from Lemma 5.2's cone construction and the explicit radial map; if the inequality fails at any small t, Theorem 5.1—and hence the existence theorem and pole singularity—would collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. The pole-type singularity of Theorem 3.3 follows from a sound comparison of the potential G_x0 with the constant-coefficient fundamental solution Φ_x0; the error v is controlled by local maximum principle/Schauder estimates and the L^p bounds of Lemma 5.8. The existence/uniqueness chain (Theorem 5.1 → Proposition 5.3 → Proposition 5.5 → weak compactness → Proposition 5.7) is coherent and has no circular dependence: Proposition 3.5 is used in Lemma 5.6 only after being derived from Theorem 5.1. The main external dependency is the stability estimate (16) quoted from (Manole et al., 2024, Theorem 6) and applied to the atomic P_t; the paper does not restate the theorem's hypotheses, so this is the single point worth re-checking, but the standard use of that theorem for empirical (atomic) Q suggests it is valid. The omitted d=2 proof of Proposition 3.5 is recoverable from the d=2 rate t^2|log t| by the same John's-lemma argument, and the Proposition 4.1(iii) Hessian/gradient typo does not enter the main proof. The regularity assumptions in Definition 2.2 are explicit; the theorems are conditional on them.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":43625,"tokens_out":19357,"duration_ms":201105,"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":[{"comment":"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.","section":"Section 5.1, Eq. (16)"},{"comment":"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.","section":"Section 5.6, Proposition 3.5"}],"minor_comments":[{"comment":"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.","section":"Section 4.1 / Proof of Proposition 4.1(iii)"},{"comment":"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.","section":"Section 5.5, Step 2.2"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"This is a strong theory paper and the core argument appears internally consistent under the stated regularity assumptions. The main reasons for major revision are completeness issues: the d=2 case of a load-bearing proposition is omitted, and the external stability theorem used at the very foundation is not stated with hypotheses. Both are fixable within the manuscript's scope. I do not see a need to question the novelty or the correctness of the main PDE derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result, and the proof strategy holds up. The paper computes the influence function of transport quantiles under Huber contamination, shows it exists away from the singular point, identifies it as the gradient of a Green-type potential for a uniformly elliptic Neumann problem, and proves the two-sided bound ||I(x0;Q_P(z))|| ≍ ||z - F_P(x0)||^{-(d-1)} in d≥2. That directly implies the influence function has infinite second moment, so the standard sqrt(n) asymptotic linear representation with a square-integrable influence function is out of reach; the authors conjecture stable non-Gaussian limits with n^{1/d} scaling. As far as I can tell, the main theorems are correct. The proof is a serious PDE argument: L2 stability, Lq bounds, Moser iteration for local L∞, Schauder estimates for C^{2,α}, then localization with a frozen fundamental solution to extract the singularity rate. The symmetry identity for the Green potential is proved, not assumed. This is the first characterization of the IF for transport quantiles in d≥2, and it resolves a natural open question.\n\nThe soft spots are modest but real. Proposition 3.5 states the shape of the preimage set K_{t,x0} in d=2, but the proof says 'the details are omitted' — that is a gap in a stated result, even if the d≥3 case is proven and the d=2 rate t^2|log t| suggests the d=2 version is recoverable. The numerical experiments are not reproducible: no code, no data-generation details, only figures and p-values. A referee should ask for a supplement. There is also a typo in Proposition 4.1(iii): the IF is ∇G, not ∇²G; it does not affect the main proofs. The paper leans on a stability estimate from Manole et al. (2024, Theorem 6) for atomic P_t; the hypotheses are not restated, but this is standard and I don't see a circularity.\n\nThe regularity assumptions (C^{2,1} supports, densities bounded away from zero, C^{1,α}) are strong but explicit; the results are conditional on them, and that's fine.\n\nBottom line: the central claim is novel and, in my reading, correct. This deserves a serious referee and is publishable after minor-to-moderate revision. I would engage with it and cite it if I worked in this area.","headline":"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.","tokens_in":44107,"tokens_out":2671,"would_cite":true,"duration_ms":29540,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G35","62G30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["transport-based quantiles","influence function","robust statistics","optimal transport","pole singularity","heavy-tailed asymptotics","empirical quantiles","fundamental-solution potential"],"falsifier":"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.","tokens_in":43225,"feed_emoji":"🎯","tokens_out":6706,"duration_ms":63738,"temperature":0.7,"pith_summary":"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.","feed_headline":"In d≥2, transport quantiles diverge on inliers","feed_subtitle":"Infinitesimal contamination at the quantile level costs a pole; the influence function fails to be square-integrable.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["In d≥2, transport quantile influence has pole singularity","Transport quantiles: influence diverges as inlier approaches","Infinite second moment for transport quantile influence","Pole-type influence: transport quantiles differ from univariate","For d≥2, quantile influence diverges at inliers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["In d≥2, transport quantile influence has pole singularity","Transport quantiles: influence diverges as inlier approaches","Infinite second moment for transport quantile influence","Pole-type influence: transport quantiles differ from univariate","For d≥2, quantile influence diverges at inliers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000222,"raw_usage":{"total_tokens":1392,"prompt_tokens":949,"completion_tokens":443,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":693,"completion_tokens_details":{"reasoning_tokens":374}},"tokens_in":693,"tokens_out":443,"duration_ms":5068,"temperature":1.0,"reasoning_tokens":374,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T13:29:14.383979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}