On the convergence of discrete dynamic unbalanced transport models

Bowen LI; Jun ZOU

doi:10.1051/m2an/2024024

On the convergence of discrete dynamic unbalanced transport models

Bowen LI^*, Jun ZOU

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

1 Citation (Scopus)

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Abstract

A generalized unbalanced optimal transport distance WB_Λ on matrix-valued measures M(Ω, Sⁿ₊) was defined in Li and Zou (arXiv:2011.05845) à la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WB_Λ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein-Fisher-Rao distance, thanks to the static formulation, we show that such an assumption can be removed. © The authors. Published by EDP Sciences, SMAI 2024.

Original language	English
Pages (from-to)	957-992
Journal	ESAIM: Mathematical Modelling and Numerical Analysis
Volume	58
Issue number	3
DOIs	https://doi.org/10.1051/m2an/2024024
Publication status	Published - May 2024
Externally published	Yes

Funding

Jun Zou was substantially supported by Hong Kong RGC General Research Fund (projects 14308322 and 14306921) and NSFC/Hong Kong RGC Joint Research Scheme 2022/23 (project N CUHK465/22).

Research Keywords

Convergence analysis
Discrete transportation metric
Unbalanced optimal transport

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This full text is made available under CC-BY 4.0. https://creativecommons.org/licenses/by/4.0/

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title = "On the convergence of discrete dynamic unbalanced transport models",

abstract = "A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures M(Ω, Sn+) was defined in Li and Zou (arXiv:2011.05845) {\`a} la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein-Fisher-Rao distance, thanks to the static formulation, we show that such an assumption can be removed. {\textcopyright} The authors. Published by EDP Sciences, SMAI 2024.",

keywords = "Convergence analysis, Discrete transportation metric, Unbalanced optimal transport",

author = "Bowen LI and Jun ZOU",

year = "2024",

month = may,

doi = "10.1051/m2an/2024024",

language = "English",

volume = "58",

pages = "957--992",

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TY - JOUR

T1 - On the convergence of discrete dynamic unbalanced transport models

AU - LI, Bowen

AU - ZOU, Jun

PY - 2024/5

Y1 - 2024/5

N2 - A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures M(Ω, Sn+) was defined in Li and Zou (arXiv:2011.05845) à la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein-Fisher-Rao distance, thanks to the static formulation, we show that such an assumption can be removed. © The authors. Published by EDP Sciences, SMAI 2024.

AB - A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures M(Ω, Sn+) was defined in Li and Zou (arXiv:2011.05845) à la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein-Fisher-Rao distance, thanks to the static formulation, we show that such an assumption can be removed. © The authors. Published by EDP Sciences, SMAI 2024.

KW - Convergence analysis

KW - Discrete transportation metric

KW - Unbalanced optimal transport

UR - http://www.scopus.com/inward/record.url?scp=85196082114&partnerID=8YFLogxK

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85196082114&origin=recordpage

U2 - 10.1051/m2an/2024024

DO - 10.1051/m2an/2024024

M3 - RGC 21 - Publication in refereed journal

SN - 2822-7840

VL - 58

SP - 957

EP - 992

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

IS - 3

ER -

On the convergence of discrete dynamic unbalanced transport models

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