Weak optimal total variation transport problems and generalized Wasserstein barycenters

TL;DR

This paper develops a duality theory for weak optimal total variation transport problems, extends classical results, and applies these to establish existence and consistency of generalized Wasserstein barycenters.

Contribution

It introduces a Kantorovich duality for weak total variation transport and applies it to prove existence and consistency of generalized Wasserstein barycenters.

Findings

Established Kantorovich duality for weak total variation transport.

Recovered duality formula for partial optimal transports.

Proved existence and consistency of generalized Wasserstein barycenters.

Abstract

In this paper, we establish a Kantorovich duality for weak optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich-Rubinstein Theorem for generalized Wasserstein distance $\widetilde{W}_1^{a,b}$ proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.