On the Private Estimation of Smooth Transport Maps

Estimating optimal transport maps between two distributions from respective samples is an important element for many machine learning methods. To do so, rather than extending discrete transport maps, it has been shown that estimating the Brenier potential of the transport problem and obtaining a transport map through its gradient is near minimax optimal for smooth problems. In this paper, we investigate the private estimation of such potentials and transport maps with respect to the distribution samples.We propose a differentially private transport map estimator with $L^2$ error at most $n^{-1} \vee n^{-\frac{2 \alpha}{2 \alpha - 2 + d}} \vee (n\epsilon)^{-\frac{2 \alpha}{2 \alpha + d}} $ up do polylog terms where $n$ is the sample size, $\epsilon$ is the desired level of privacy, $\alpha$ is the smoothness of the true transport map, and $d$ is the dimension of the feature space. We also provide a lower bound for the problem.

Imagine you need to rearrange piles of sand from one shape to another using the least effort possible—this is the essence of optimal transport, a powerful concept used in fields like machine learning, economics, and data analysis. When we only have a handful of sand grains (samples) to work with, and those grains represent sensitive information—like personal data—we must protect privacy while still finding an efficient way to move the sand. This paper introduces a new method to tackle this challenge, ensuring both accuracy in rearranging the sand and strong privacy safeguards for the data, paving the way for safer and smarter data-driven solutions.