Distribution-on-Distribution Regression via Optimal Transport Maps

TL;DR

This paper introduces a novel regression framework for probability distributions using optimal transport theory, establishing a consistent estimator that simplifies to isotonic regression, with applications demonstrated on real and simulated data.

Contribution

It proposes a new distribution-on-distribution regression model based on optimal transport maps, with a consistent estimator and practical computation method.

Findings

Estimator is consistent and converges at a quantifiable rate.

Regression reduces to an isotonic regression problem, enabling easy implementation.

Method is validated on both real and simulated datasets.

Abstract

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. We illustrate our methodology using real and simulated data.