Multi-transport Distributional Regression

TL;DR

This paper introduces a novel Wasserstein-based regression framework for modeling how multiple distributional predictors influence a response distribution, providing interpretability and theoretical guarantees.

Contribution

It proposes an intrinsic regression model that handles multiple distributional predictors via a weighted Fréchet mean in Wasserstein space, with theoretical identifiability and asymptotic properties.

Findings

Improved predictive performance over existing methods

Model provides interpretable weights for predictors

Theoretical guarantees for estimator consistency

Abstract

We study distribution-on-distribution regression problems in which a response distribution depends on multiple distributional predictors. Such settings arise naturally in applications where the outcome distribution is driven by several heterogeneous distributional sources, yet remain challenging due to the nonlinear geometry of the Wasserstein space. We propose an intrinsic regression framework that aggregates predictor-specific transported distributions through a weighted Fr\'echet mean in the Wasserstein space. The resulting model admits multiple distributional predictors, assigns interpretable weights quantifying their relative contributions, and defines a flexible regression operator that is invariant to auxiliary construction choices, such as the selection of a reference distribution. From a theoretical perspective, we establish identifiability of the induced regression operator and derive asymptotic guarantees for its estimation under a predictive Wasserstein semi-norm, which directly characterizes convergence of the composite prediction map. Extensive simulation studies and a real data application demonstrate the improved predictive performance and interpretability of the proposed approach compared with existing Wasserstein regression methods.