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Unbalanced optimal total variation transport problems and generalized Wasserstein barycenters

Part of: Existence theories Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  04 June 2021

Nhan-Phu Chung  and
Thanh-Son Trinh
Nhan-Phu Chung
Affiliation:
Department of Mathematics, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 16419, Korea (phuchung@skku.edu; phuchung82@gmail.com; sontrinh@skku.edu)
Thanh-Son Trinh
Affiliation:
Department of Mathematics, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 16419, Korea (phuchung@skku.edu; phuchung82@gmail.com; sontrinh@skku.edu)
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Abstract

In this paper, we establish a Kantorovich duality for unbalanced 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.

Keywords

Unbalanced optimal transportdualitygeneralized Wasserstein spacebarycenter

MSC classification

Primary: 90C46: Optimality conditions, duality
Secondary: 49J40: Variational methods including variational inequalities 58E30: Variational principles
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 674 - 700
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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