Distribution-on-Distribution Regression with Wasserstein Metric: Multivariate Gaussian Case

TL;DR

This paper introduces a novel regression model for mapping between Gaussian distributions using Wasserstein geometry, providing an interpretable framework with proven convergence and superior performance in simulations and weather data applications.

Contribution

The study develops a Wasserstein-based Gaussian distribution regression model that leverages the geometry of Wasserstein space for simplicity and interpretability, extending to non-Gaussian cases.

Findings

Models outperform alternative methods in simulations

Convergence rates of prediction errors are established

Application to weather data demonstrates practical utility

Abstract

Distribution data refers to a data set where each sample is represented as a probability distribution, a subject area receiving burgeoning interest in the field of statistics. Although several studies have developed distribution-to-distribution regression models for univariate variables, the multivariate scenario remains under-explored due to technical complexities. In this study, we introduce models for regression from one Gaussian distribution to another, utilizing the Wasserstein metric. These models are constructed using the geometry of the Wasserstein space, which enables the transformation of Gaussian distributions into components of a linear matrix space. Owing to their linear regression frameworks, our models are intuitively understandable, and their implementation is simplified because of the optimal transport problem's analytical solution between Gaussian distributions. We also explore a generalization of our models to encompass non-Gaussian scenarios. We establish the convergence rates of in-sample prediction errors for the empirical risk minimizations in our models. In comparative simulation experiments, our models demonstrate superior performance over a simpler alternative method that transforms Gaussian distributions into matrices. We present an application of our methodology using weather data for illustration purposes.