  We develop a computationally tractable method for estimating the optimal map
between two distributions over $\mathbb{R}^d$ with rigorous finite-sample
guarantees. Leveraging an entropic version of Brenier's theorem, we show that
our estimator -- the \emph{barycentric projection} of the optimal entropic plan
-- is easy to compute using Sinkhorn's algorithm. As a result, unlike current
approaches for map estimation, which are slow to evaluate when the dimension or
number of samples is large, our approach is parallelizable and extremely
efficient even for massive data sets. 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. We showcase the efficacy of our proposed estimator through
numerical examples, even ones not explicitly covered by our assumptions. By
virtue of Lepski's method, we propose a modified version of our estimator that
is adaptive to the smoothness of the underlying optimal transport map. Our
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).
