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arXiv · 2606.04683

Minimax Private Estimation of Smooth Optimal-Transport Maps

Clément Lalanne·David Rodríguez-Vítores·Franck Iutzeler·Jean-Michel Loubes
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Abstract

We study the problem of estimating smooth optimal transport (OT) maps between two probability distributions under differential privacy (DP) constraints. Leveraging wavelet-based density estimators and recent stability bounds for smooth OT maps, we propose differentially private estimators that apply to both central and local DP models. Our main estimator achieves near-minimax optimal rates in dimension $d \geq 2$, and we complement it with a quantile-based estimator that attains minimax optimal rates in dimension $d = 1$ under central DP. We further establish matching minimax lower bounds, confirming the near-optimality of our approach. To the best of our knowledge, this constitutes the first differentially private procedure for OT map estimation with minimax optimality guarantees.

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Clément Lalanne, David Rodríguez-Vítores, Franck Iutzeler, Jean-Michel Loubes. 2026-06-03. Minimax Private Estimation of Smooth Optimal-Transport Maps. https://arxiv.org/abs/2606.04683

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