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arXiv · 2301.07934

Weak log-majorization between the geometric and Wasserstein means

Luyining Gan·Sejong Kim
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Abstract

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.

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Luyining Gan, Sejong Kim. 2023-01-19. Weak log-majorization between the geometric and Wasserstein means. https://arxiv.org/abs/2301.07934

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