Quantitative Stability of the Shadow for Wasserstein Projections and Sample Complexity

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-Hölder 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 Hölder continuity of optimal transport maps established by \citet{Quantitative_stability_duke2023}, together with its recent extension by \citet{mischler2025quantitativestabilityoptimaltransport}.

• The paper establishes the bi-Hölder continuity of the shadow mapping under Wasserstein projections, providing explicit stability bounds.
• It leverages contraction properties of Lᵖ-projections and the stability of optimal transport maps to address the complexity of multimarginal optimal transport.
• The sample complexity analysis delivers robust convergence rates for empirical shadows, underpinning practical applications in statistical optimal transport.

Quantitative Stability and Sample Complexity of the Shadow for Wasserstein Projections

Introduction

The paper "Quantitative Stability of the Shadow for Wasserstein Projections and Sample Complexity" (2604.17711) investigates the quantitative stability properties of the shadow, a specialized projection onto the set of couplings with given marginals under the Wasserstein distance. The shadow concept, introduced by Eckstein and Nutz, is utilized to understand the stability of algorithms such as Sinkhorn and has growing importance in statistical settings, particularly when extending couplings in multimarginal contexts. The research establishes bi-Hölder continuity properties for the shadow mapping and derives its sample complexity under mild regularity conditions, leveraging recent advances in optimal transport map stability.

The Shadow Projection and Wasserstein Projection Problem

Given a measure $\rho$ and a vector of marginals $\bm{\mu}$, the Wasserstein projection aims to find a coupling $\pi$ with marginals $\bm{\mu}$ that minimizes $W_p(\pi, \rho)$, which is equivalent to a specific multimarginal optimal transport (MOT) problem with separable cost. The computational complexity of generic MOT is NP-hard as the number of marginals $K$ grows, but the separable structure in this setting allows tractability.

The shadow projection, as characterized in [Eckstein_Nutz_2022], comprises composing the optimal couplings for each marginal using corresponding kernels, thereby defining $\mathcal{S}(\rho; \bm{\mu})$. While the projection problem may admit multiple solutions, under absolute continuity and suitable regularity, the shadow is unique and obtained via optimal transport maps between marginals. This uniqueness is crucial for rigorous quantitative analysis.

Main Stability Results

Bi-Hölder Continuity

The core result is the bi-Hölder continuity of the shadow map. Let $\rho, \xi$ be absolutely continuous measures and $\bm{\mu}, \bm{\nu}$ their respective marginal vectors. The Wasserstein distance between their shadows satisfies:

$$W_p(\bm{\mu}, \bm{\nu}) \leq W_p(\mathcal{S}(\rho; \bm{\mu}), \mathcal{S}(\xi; \bm{\nu})) \leq W_p(\rho, \xi) + C \sum_{i=1}^K W_q(\mu_i, \nu_i)^{\theta(p)}$$

where $\bm{\mu}$0 is determined by the regularity properties of the optimal transport maps (e.g., $\bm{\mu}$1 for $\bm{\mu}$2; $\bm{\mu}$3 for $\bm{\mu}$4), and $\bm{\mu}$5 is independent of the marginals.

The proof combines two techniques:

• A contraction property for the $\bm{\mu}$6-projection developed in recent works [kim2025stabilitywassersteinprojectionsconvex, alfonsi2025wassersteinprojectionsconvexorder].
• Hölder stability of optimal transport maps, leveraging the results for $\bm{\mu}$7-Wasserstein by [Quantitative_stability_duke2023] and its extensions to $\bm{\mu}$8 in [mischler2025quantitativestabilityoptimaltransport].

Relaxing Regularity Conditions

By smoothing empirical or irregular distributions—convolving with absolutely continuous kernels—the absolute continuity requirement can be removed for one of the measures (but not both). The triangle inequality allows quantitative stability bounds for shadows of smoothed measures, establishing existence (not always uniqueness) of shadows with desirable stability properties.

Sample Complexity and Statistical Optimal Transport

The paper exploits the stability bounds to analyze sample complexity in empirical settings. When both the base measure and marginals are replaced with empirical distributions, the Wasserstein distance between empirical shadows can be bounded above using convergence rates of the empirical measures (as per intrinsic Wasserstein dimension), plus a Hölder term for the marginals:

$\bm{\mu}$9

where $\pi$0 and $\pi$1 are sample sizes for the base and marginal empirical distributions, $\pi$2 is the ambient dimension, and $\pi$3 are Wasserstein dimensions reflecting the intrinsic complexity of support [JW_FB_sample_rates].

Implications and Future Directions

The established bi-Hölder continuity for the shadow under Wasserstein projections has several implications:

• Provides rigorous quantitative guarantees on projection stability as marginals and source distributions are perturbed.
• Enables sample size analysis for statistical optimal transport methods in multimarginal settings, impacting coupling extensions in empirical data, effective sample scaling, and algorithmic robustness.
• Offers insights for practical implementations in Sinkhorn-based algorithms, where shadow projections govern stability under regularization.
• Raises open theoretical questions regarding stability properties for $\pi$4, necessitating new frameworks beyond reverse Poincaré inequalities.

Future developments may include:

• Extending the stability framework to higher-order Wasserstein distances ($\pi$5) and less restrictive assumptions.
• Incorporating more general multimarginal cost functions or constraints, relevant in martingale OT or weak OT areas.
• Tightening sample complexity bounds by exploiting finer geometric or regularity properties of support.

Conclusion

This work rigorously quantifies the stability and sample complexity of the shadow for Wasserstein projections, leveraging bi-Hölder continuity and contraction properties of $\pi$6 projections. Its theoretical and statistical implications enable stable coupling projections and facilitate practical analysis of empirical transport, laying the foundation for further advances in multimarginal optimal transport and statistical learning under Wasserstein-projected couplings.