Kantorovich Regression Analysis of Random Distributions with Mixed Predictors

TL;DR

This paper introduces a novel Kantorovich regression model for analyzing distribution-valued responses with mixed predictors, leveraging optimal transport theory to interpret and predict complex distributional data.

Contribution

The paper develops a new regression framework combining Wasserstein barycenters and Kantorovich potentials, with theoretical guarantees and practical algorithms for mixed distributional and Euclidean predictors.

Findings

Successfully applied to housing price distribution data.

Effectively analyzed temperature distribution data.

Demonstrated model's interpretability and flexibility.

Abstract

We study regression problems with distribution-valued responses and mixed distributional and Euclidean predictors. In quadratic cost, the negative gradient of the Kantorovich potential represents, at each source location, the displacement to its matched location under the optimal transport map. By constructing potentials from the Wasserstein barycenter to individual distributions, the proposed Kantorovich regression model approximates the response displacement field as a sum of predictor displacement fields, each adjusted by a functional parameter. Owing to the linear structure, Euclidean predictors can enter as scaling coefficients of $c$-concave parameter potentials. We characterize functional parameter classes ensuring the intrinsic structure of the model, establish asymptotic theory through uniform convergence of the empirical Wasserstein loss, and derive G\^ateaux derivatives leading to first-order optimization algorithms. Real data applications include a mixed-predictor analysis of housing price distributions and an analysis of two-dimensional temperature distributions, demonstrating the flexibility and interpretability of the proposed framework.