Weak log-majorization between the geometric and Wasserstein means

There exist lots of distinct geometric means on the cone of positive definite Hermitian matrices such as the metric geometric mean, spectral geometric mean, log-Euclidean mean and Wasserstein mean. In this paper, we prove the log-majorization relation on the singular values of the product of given t...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2024-02, Vol.530 (2), p.127711, Article 127711
Main Authors: Gan, Luyining, Kim, Sejong
Format: Article
Language:English
Subjects:
Metric geometric mean
Positive definite matrix
Spectral geometric mean
Wasserstein mean
Weak log-majorization
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Summary:There exist lots of distinct geometric means on the cone of positive definite Hermitian matrices such as the metric geometric mean, spectral geometric mean, log-Euclidean mean and Wasserstein mean. In this paper, we prove the log-majorization relation on the singular values of the product of given two positive definite matrices and their (metric and spectral) geometric means. We also establish the weak log-majorization between the spectra of two-variable Wasserstein mean and spectral geometric mean. In particular, we verify with certain condition on variables that two-variable Wasserstein mean converges decreasingly to the log-Euclidean mean with respect to the weak log-majorization.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2023.127711