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Wasserstein barycenters of compactly supported measures

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Abstract

We consider in this paper probability measures with compact support on the open convex cone of positive definite Hermitian matrices. We define the least squares barycenter for the Bures–Wasserstein distance, called the Wasserstein barycenter, as a unique minimizer of the integral of squared Bures–Wasserstein distances. Furthermore, interesting properties of the Wasserstein barycenter including the iteration approach via optimal transport maps, the boundedness and extended Lie–Trotter–Kato formula, and several inequalities with power means have been established in the setting of compactly supported measures.

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Acknowledgements

The work of Sejong Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1C1B60 01394). The work of Hosoo Lee was supported by the 2020 scientific promotion program funded by Jeju National University, and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1D1A1B0704 9948)

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Authors and Affiliations

  1. Department of Mathematics, Chungbuk National University, Cheongju, 28644, Korea

    Sejong Kim

  2. Elementary Education Research Institute, Jeju National University, Jeju, 63294, Korea

    Hosoo Lee

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  1. Sejong Kim
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  2. Hosoo Lee
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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Hosoo Lee.

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Kim, S., Lee, H. Wasserstein barycenters of compactly supported measures. Anal.Math.Phys. 11, 153 (2021). https://doi.org/10.1007/s13324-021-00588-z

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  • DOI: https://doi.org/10.1007/s13324-021-00588-z

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