Statistical estimation of Monge transport maps with empirical Brenier potentials

par Edouard Pauwels (TSE)

mardi 14 avr. 2026, 11:15 → 12:15 Europe/Paris

Salle K. Johnson (1R3, 1er étage)

We analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem. For absolutely continuous source measures, this map is uniquely defined as the gradient of a convex function, a result known as Brenier 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: a Brenier potential. This characterization is the basis for our Monge transport map estimator, for measures known only through finite samples. The resulting Brenier potential has a simple closed form expression based on the dual solution of the discrete sampled problem. We exhibit convergence rates for this estimator based on a new error bound for the quadratic optimal transport problem. Our methodology does not rely on smoothness or continuity of the Monge transport map and requires no computation beyond primal-dual solutions of the discrete finite dimensional linear program transport problem.