Fr\'echet regression of multivariate distributions with nonparanormal transport

TL;DR

This paper introduces a novel regression method for multivariate distributional responses using the nonparanormal transport metric, enabling efficient estimation and interpretation while addressing computational and statistical challenges.

Contribution

It develops a new regression framework that models multivariate distributions within the nonparanormal family using NPT, providing theoretical guarantees and practical advantages.

Findings

NPT is topologically equivalent to Wasserstein distance.

The method mitigates the curse of dimensionality.

Achieves fast convergence rates in estimation.

Abstract

Regression with distribution-valued responses and Euclidean predictors has gained increasing scientific relevance. While methodology for univariate distributional data has advanced rapidly in recent years, multivariate distributions, which additionally encode dependence across univariate marginals, have received less attention and pose computational and statistical challenges. In this work, we address these challenges with a new regression approach for multivariate distributional responses, in which distributions are modeled within the semiparametric nonparanormal family. By incorporating the nonparanormal transport (NPT) metric -- an efficient closed-form surrogate for the Wasserstein distance -- into the Fr\'echet regression framework, our approach decomposes the problem into separate regressions of marginal distributions and their dependence structure, facilitating both efficient estimation and granular interpretation of predictor effects. We provide theoretical justification for NPT, establishing its topological equivalence to the Wasserstein distance and proving that it mitigates the curse of dimensionality. We further prove uniform convergence guarantees for regression estimators, both when distributional responses are fully observed and when they are estimated from empirical samples, attaining fast convergence rates comparable to the univariate case. The utility of our method is demonstrated via simulations and an application to continuous glucose monitoring data.