Minimax Rate for Optimal Transport Regression Between Distributions

TL;DR

This paper establishes the optimal rate of estimation for distribution-to-distribution regression models based on optimal transport, providing theoretical guarantees for the Fréchet least squares estimator.

Contribution

It derives the minimax rate for estimating the regression function in distribution-on-distribution regression using optimal transport, matching the estimator's convergence rate.

Findings

The minimax lower bound matches the convergence rate of the estimator.

The Fréchet least squares estimator attains the optimal rate.

Provides theoretical foundation for distributional regression using optimal transport.

Abstract

Distribution-on-distribution regression considers the problem of formulating and estimating a regression relationship where both covariate and response are probability distributions. The optimal transport distributional regression model postulates that the conditional Fr\'echet 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 Fr\'echet least squares estimator.