Transportation of Measure Regression in Higher Dimensions

TL;DR

This paper introduces a higher-dimensional optimal transport regression framework for probability distributions, extending existing models to Gaussian and compactly supported distributions, with theoretical convergence guarantees and an efficient computational algorithm.

Contribution

It generalizes transportation-based regression to higher dimensions and Gaussian distributions, providing convergence rates and an iterative DC programming algorithm for efficient estimation.

Findings

Estimator achieves parametric convergence rate in Gaussian case

Algorithm simplifies to finite-dimensional optimization

Demonstrated effective performance in simulations

Abstract

We present an optimal transport framework for performing regression when both the covariate and the response are probability distributions on a compact Euclidean subset $\Omega\subset\mathbb{R}^d$, where $d>1$. Extending beyond compactly supported distributions, this method also applies when both the predictor and responses are Gaussian distributions on $\mathbb{R}^d$. Our approach generalizes an existing transportation-based regression model to higher dimensions. This 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 an upper bound for the rate of convergence of a plug-in estimator. We propose an iterative algorithm for computing the estimator, which is based on DC (Difference of Convex Functions) Programming. In the Gaussian case, the estimator achieves a parametric rate of convergence, and the computation of the estimator simplifies to a finite-dimensional optimization over positive definite matrices, allowing for an efficient solution. The performance of the estimator is demonstrated in a simulation study.