Estimation Rates for Optimal Transport Maps to the Barycenter in Multidimensional Spaces

The optimal transport barycenter problem seeks transport maps that align multiple distributions with their barycenter, a central object in applications such as image interpolation and fair regression. Although several algorithms estimate these maps in continuous multidimensional spaces, their statistical estimation rates remain largely unknown. We establish, to our knowledge, the first convergence rate for optimal transport map estimation to the barycenter under the quadratic cost in multidimensional Euclidean spaces. Crucially, the rate is governed solely by the approximation error of the estimator's class, possibly non-convexly parametrized such as deep neural networks, and requires no regularity of the underlying transport maps. The resulting rate interpolates between the fast rate under the Poincar\'e inequality and the slow rate in general, two extremes previously established only in isolation for transport map estimation between two distributions.