Generalized Wasserstein barycenters between probability measures living on different subspaces

TL;DR

This paper extends Wasserstein barycenters to probability measures on different subspaces, exploring theoretical properties, dual formulations, and numerical computation methods, including explicit solutions for Gaussian cases.

Contribution

It introduces a novel generalization of Wasserstein barycenters for measures on different subspaces, with theoretical analysis and practical algorithms.

Findings

Existence and uniqueness of the generalized barycenter

Connection to multi-marginal optimal transport

Explicit solution for Gaussian distributions

Abstract

In this paper, we introduce a generalization of the Wasserstein barycenter, to a case where the initial probability measures live on different subspaces of R^d. We study the existence and uniqueness of this barycenter, we show how it is related to a larger multi-marginal optimal transport problem, and we propose a dual formulation. Finally, we explain how to compute numerically this generalized barycenter on discrete distributions, and we propose an explicit solution for Gaussian distributions.