Distribution-on-Distribution Regression via Optimal Transport Maps

Understand

We present a framework for performing regression when both covariate and response are probability distributions on a compact interval $\Omega\subset\mathbb{R}$.

• Our regression model is based on the theory of optimal transportation and links the conditional Fr\'echet mean of the response distribution to the covariate distribution via an optimal transport map.
• We define a Fr\'echet-least-squares estimator of this regression map, and establish its consistency and rate of convergence to the true map, under both full and partial observation of the regression pairs.
• Computation of the estimator is shown to reduce to an isotonic regression problem, and thus our regression model can be implemented with ease.