Regression analysis of distributional data through Multi-Marginal Optimal transport

TL;DR

This paper introduces a novel regression framework for distributional data in Wasserstein space, enabling modeling of complex temporal distributional changes via multi-marginal optimal transport.

Contribution

It formulates a regression problem in Wasserstein space using multi-marginal optimal transport, allowing flexible curve modeling for distributional data.

Findings

Efficient computational method for Wasserstein regression.

Applicable to diverse data types like images and spectra.

Demonstrated with academic examples.

Abstract

We formulate and solve a regression problem with time-stamped distributional data. Distributions are considered as points in the Wasserstein space of probability measures, metrized by the 2-Wasserstein metric, and may represent images, power spectra, point clouds of particles, and so on. The regression seeks a curve in the Wasserstein space that passes closest to the dataset. Our regression problem allows utilizing general curves in a Euclidean setting (linear, quadratic, sinusoidal, and so on), lifted to corresponding measure-valued curves in the Wasserstein space. It can be cast as a multi-marginal optimal transport problem that allows efficient computation. Illustrative academic examples are presented.