Minimax rate for optimal transport regression between distributions

Distribution-on-distribution regression considers the problem of formulating and es-timating a regression relationship where both covariate and response are probability distributions. The optimal transport distributional regression model postulates that the conditional Frechet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish the minimax rate of estimation of such a regression function, by deriving a lower bound that matches the convergence rate attained by the Frechet least squares estimator.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).