In this talk, I will be talking about a computationally tractable method for estimating the optimal map between two distributions. We  leverage an entropic version of Brenier’s theorem and show that our estimator—the barycentric projection of the optimal entropic plan—is easy to compute using Sinkhorn’s algorithm. Under smoothness assumptions on the optimal map, we show that our estimator enjoys comparable statistical performance  to other estimators in the literature, but with much lower computational cost. The proofs are based on a modified duality principle for entropic optimal transport and on a method for approximating optimal entropic plans due to Pal (2019).