Remove the Poincaré assumption for fast transport map estimation rates

Determine whether the fast L2(µ) error rates proved for the semidual empirical risk minimizer of the optimal transport map between probability measures on R^d can be achieved without assuming that the source measure µ satisfies a Poincaré inequality. Concretely, establish fast convergence rates for estimating the Brenier map from µ to ν under milder or no functional inequality assumptions on µ.

Background

The chapter develops a semidual empirical risk minimization approach for estimating optimal transport maps and derives fast convergence rates by localizing the estimator and applying a refined chaining bound. A key assumption used in the fast-rate analysis is that the source measure µ satisfies a Poincaré inequality, which links L2 and gradient norms and enables localization.

The authors note that this functional inequality may be restrictive and raise the question of whether the same fast rates can be obtained without it, pointing out that removing the assumption remains unresolved.