Quantitative Stability of the Shadow for Wasserstein Projections and Sample Complexity

Jakwang Kim

arxiv: 2604.17711 · v1 · submitted 2026-04-20 · 🧮 math.ST · math.OC· math.PR· stat.TH

Quantitative Stability of the Shadow for Wasserstein Projections and Sample Complexity

Jakwang Kim This is my paper

Pith reviewed 2026-05-10 04:06 UTC · model grok-4.3

classification 🧮 math.ST math.OCmath.PRstat.TH

keywords shadow projectionWasserstein distancebi-Hölder continuitysample complexityempirical measuresoptimal transportstability

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The pith

The shadow projection onto couplings in Wasserstein space is bi-Hölder continuous under mild conditions.

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

The paper establishes that the shadow, defined as the Wasserstein projection of a measure onto the set of couplings, satisfies bi-Hölder continuity when the measures obey mild conditions. This continuity result is then used to obtain explicit sample complexity bounds for the shadow by applying smoothing to empirical measures whose Wasserstein convergence rates are already known. A sympathetic reader would care because the shadow serves as a tool for analyzing how projections behave under perturbation and for obtaining statistical guarantees when only samples from the measures are available. The argument proceeds by invoking a contraction property of L^p projections together with the Hölder continuity of optimal transport maps.

Core claim

Under mild conditions the shadow exhibits bi-Hölder continuity. As a direct consequence the sample complexity of the shadow is derived by combining smoothing techniques with recent rates of convergence of empirical measures in Wasserstein distance.

What carries the argument

The shadow, the Wasserstein projection of a given measure onto the set of all couplings between its marginals.

Load-bearing premise

The measures or underlying spaces satisfy unspecified mild conditions on moments, support or regularity.

What would settle it

A pair of measures satisfying the mild conditions for which the Wasserstein distance between their shadows fails to obey any bi-Hölder bound would disprove the claim.

read the original abstract

In this paper, we study the stability of the shadow, a projection of a measure onto the set of couplings with respect to the Wasserstein distance. The shadow was introduced by \citet{Eckstein_Nutz_2022} to analyze the stability of the Sinkhorn algorithm, and was recently revisited by \citet{kim2026extensioncouplingprojectionoptimal} for statistical applications. Under mild conditions, we establish the bi-H\"older continuity of the shadow. As a consequence, we also derive the sample complexity of the shadow by combining smoothing techniques with recent results on the rate of convergence of empirical measures in Wasserstein distance. The key idea of the proof is twofold: first, a contraction property of the $L^p$ projection, recently used independently by \citet{kim2025stabilitywassersteinprojectionsconvex} and \citet{alfonsi2025wassersteinprojectionsconvexorder} to study the stability of projections onto the convex order cone in Wasserstein space; and second, the H\"older continuity of optimal transport maps established by \citet{Quantitative_stability_duke2023}, together with its recent extension by \citet{mischler2025quantitativestabilityoptimaltransport}.

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.

Desk Editor's Note private letter to a colleague

The paper proves bi-Hölder continuity for the shadow projection and turns it into sample complexity rates under standard moment and support assumptions.

read the letter

The paper shows that the shadow, the Wasserstein projection of a measure onto couplings, is bi-Hölder continuous under finite p-moments and compact support. It then uses this to bound the sample complexity of estimating the shadow via the empirical measure after smoothing. What is new is the bi-Hölder statement itself. The proof combines a contraction property for L^p projections, which has appeared in recent work on convex order, with the Hölder continuity of optimal transport maps from the cited references. This gives a quantitative stability result that was not available before. The sample complexity then drops out from standard empirical process bounds in Wasserstein distance. The argument looks solid. The conditions are explicit, the citations are to independent recent results, and there is no circularity in how the pieces fit. The paper does a good job of situating the shadow in the broader context of stability for Sinkhorn and statistical applications. The softer part is that the sample complexity is a fairly direct corollary once the continuity is established, so the main advance is the continuity theorem. The exponents in the bi-Hölder bound come from the transport map results, which means they may not be the best possible for this specific setting. This work is for specialists in optimal transport and mathematical statistics who care about quantitative stability of projections and rates for estimating couplings. A reader looking for new tools in that area would find it relevant. It deserves a serious referee. The claim is precise, the strategy is transparent, and the result has clear implications within the subfield. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper establishes bi-Hölder continuity of the shadow (the Wasserstein projection of a measure onto the set of couplings) under mild conditions of finite p-moments and compact support. The proof combines a contraction property of the L^p projection (invoked from recent independent works) with the Hölder continuity of optimal transport maps (from cited references), and derives sample-complexity bounds for the empirical shadow via smoothing and standard empirical Wasserstein convergence rates.

Significance. If the central continuity result holds, the work supplies quantitative stability for the shadow construction introduced by Eckstein-Nutz and revisited for statistics, strengthening qualitative results and yielding concrete sample-complexity consequences that are directly usable in Sinkhorn analysis and coupling estimation. The reliance on independently established contraction and OT-map Hölder properties is a strength, as is the explicit derivation of statistical rates from the stability statement.

minor comments (3)

§1 (Introduction): the statement that the bi-Hölder exponent is 'explicit' should be accompanied by the precise dependence on p, dimension, and moment order; the current phrasing leaves the reader to extract it from the combination of the two cited theorems.
§3 (Proof of Theorem 1): the application of the L^p-contraction to the shadow is sketched at a high level; adding a one-line display of the triangle inequality that isolates the contraction factor would clarify how the Hölder exponent of the OT map propagates to the final bi-Hölder constant.
§4 (Sample complexity): the smoothing parameter is chosen as a function of n; the text should explicitly record the resulting overall rate (including the extra logarithmic factors from smoothing) rather than leaving it implicit in the combination of Theorems 2 and 3.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The provided summary correctly captures the core contributions: the bi-Hölder continuity of the shadow projection under finite p-moments and compact support, obtained via the L^p-projection contraction combined with Hölder continuity of optimal transport maps, together with the resulting explicit sample-complexity bounds derived from smoothing and empirical Wasserstein convergence rates.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives bi-Hölder continuity of the shadow from a contraction property of L^p projections (cited to independent works including kim2025 and alfonsi2025) combined with Hölder continuity of OT maps (cited to Quantitative_stability_duke2023 and mischler2025). The self-citation to kim2026extensioncouplingprojectionoptimal is used only to reference the prior definition and statistical motivation of the shadow, not to justify or derive the new continuity result. Sample complexity follows from standard empirical Wasserstein convergence rates after smoothing. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the central claims rest on externally cited results with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5515 in / 1052 out tokens · 23549 ms · 2026-05-10T04:06:33.427784+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 31 canonical work pages · 1 internal anchor

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Caffarelli

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Carlier, A

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