A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

S. Kum; Manh Hong Duong; Y. Lim; S. Yun

doi:10.1016/j.cam.2022.114588

A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

S. Kum, Manh Hong Duong, Y. Lim, S. Yun

Mathematics

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Abstract

In this paper, we focus on the analysis of the regularized Wasserstein barycenter problem. We provide uniqueness and a characterization of the barycenter for two important classes of probability measures, each regularized by a particular entropy functional: (i) Gaussian distributions and (ii) q-Gaussian distributions. We propose an algorithm based on gradient projection method (GPM) in the space of matrices in order to compute these regularized barycenters. Finally, we numerically show the influence of parameters and stability of the algorithm under small perturbation of data and compare the gradient projection method with Riemannian gradient method.

Original language	English
Article number	114588
Journal	Journal of Computational and Applied Mathematics
Volume	416
Early online date	12 Jul 2022
DOIs	https://doi.org/10.1016/j.cam.2022.114588
Publication status	Published - 21 Jul 2022

Bibliographical note

Funding Information:
The work of M. H. Duong was supported by EPSRC, UK Grants EP/W008041/1 and EP/V038516/1 . The work of S. Yun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2016R1A5A1008055 and No. 2022R1A2C1011503 ). The work of Y. Lim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) No. 2015R1A3A2031159 .

Publisher Copyright:
© 2022 The Author(s)

Keywords

Wasserstein barycenter
q-Gaussian measures
Gradient projection method
Optimization

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10.1016/j.cam.2022.114588Licence: Creative Commons: Attribution (CC BY)

KumS2022GPMbasedFinal published version, 542 KBLicence: Creative Commons: Attribution (CC BY)

Rigorous coarse-graining of defects at positive temperature
Duong, H. (Principal Investigator)
Engineering & Physical Science Research Council
1/06/22 → 31/05/23
Project: Research Councils
Variational structures, convergence to equilibrium and multiscale analysis for non-Markovian systems
Duong, H. (Principal Investigator)
Engineering & Physical Science Research Council
1/02/22 → 30/06/24
Project: Research Councils

Cite this

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title = "A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures",

abstract = "In this paper, we focus on the analysis of the regularized Wasserstein barycenter problem. We provide uniqueness and a characterization of the barycenter for two important classes of probability measures, each regularized by a particular entropy functional: (i) Gaussian distributions and (ii) q-Gaussian distributions. We propose an algorithm based on gradient projection method (GPM) in the space of matrices in order to compute these regularized barycenters. Finally, we numerically show the influence of parameters and stability of the algorithm under small perturbation of data and compare the gradient projection method with Riemannian gradient method.",

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AB - In this paper, we focus on the analysis of the regularized Wasserstein barycenter problem. We provide uniqueness and a characterization of the barycenter for two important classes of probability measures, each regularized by a particular entropy functional: (i) Gaussian distributions and (ii) q-Gaussian distributions. We propose an algorithm based on gradient projection method (GPM) in the space of matrices in order to compute these regularized barycenters. Finally, we numerically show the influence of parameters and stability of the algorithm under small perturbation of data and compare the gradient projection method with Riemannian gradient method.

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KW - Optimization

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JF - Journal of Computational and Applied Mathematics

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A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

Abstract

Bibliographical note

Keywords

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Projects

Rigorous coarse-graining of defects at positive temperature

Variational structures, convergence to equilibrium and multiscale analysis for non-Markovian systems

Cite this