Statistical Estimation of Monge Transport Maps via Brenier Potentials
We introduce and analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem in the Euclidean space. For absolutely continuous source measures, this map is uniquely defined as the gradient of a convex function, a result known as Brenier's theorem. Without absolute continuity, the problem is relaxed, maps are replaced by coupling measures, and optimal couplings are supported on the subdifferential of a convex function, called a Brenier potential. This potential is the basis for our transport map statistical estimator, for measures known only through finite samples. We construct an explicit optimal Brenier potential for the empirical problem, with a simple closed form expression based on dual solutions. Our transport map estimator is then defined as a suitable subdifferential selection, and can be extended to $\mathbb{R}^d$. We exhibit convergence rates for this estimator based on a new error bound for the quadratic optimal transport problem. In the semi-discrete setting, where the target measure is finitely supported, our estimator enjoys sharper convergence rates. Our methodology does not rely on smoothness or continuity of the target Monge transport map and requires no computation beyond primal-dual solutions of the empirical finite dimensional problem. Finally, using similar proof techniques, we provide novel convergence rates for empirical couplings.