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-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}.