ADOPT: Additive Optimal Transport Regression

TL;DR

This paper introduces ADOPT, a novel regression framework using additive optimal transport in general metric spaces, enabling flexible modeling of complex responses like distributions and matrices.

Contribution

It extends optimal transport to additive regression in general metric spaces, broadening applicability to diverse data types such as distributions and SPD matrices.

Findings

Successfully models probability distributions and SPD matrices.

Applied to fMRI data for brain imaging analysis.

Demonstrates flexibility across various metric space responses.

Abstract

Regression analysis for responses taking values in general metric spaces has received increasing attention, particularly for settings with Euclidean predictors $X \in \mathbb{R}^p$ and non-Euclidean responses $Y$ in metric spaces. While additive regression is a powerful tool for enhancing interpretability and mitigating the curse of dimensionality in the presence of multivariate predictors, its direct extension is hindered by the absence of vector space operations in general metric spaces. We propose a novel framework for additive optimal transport regression, which incorporates additive structure through optimal geodesic transports. A key idea is to extend the notion of optimal transports in Wasserstein spaces to general geodesic metric spaces. This unified approach accommodates a wide range of responses, including probability distributions, symmetric positive definite (SPD) matrices with various metrics and spherical data. The practical utility of the method is illustrated with correlation matrices derived from resting state fMRI brain imaging data.