Weak log-majorization between the geometric and Wasserstein means

TL;DR

This paper establishes log-majorization relations among various geometric means of positive definite matrices, revealing how the Wasserstein mean compares to spectral geometric and log-Euclidean means.

Contribution

It proves new log-majorization relations between the spectra of different matrix means, including the Wasserstein and geometric means, and shows convergence properties under certain conditions.

Findings

Log-majorization relation between singular values and geometric means.

Weak log-majorization between Wasserstein mean and spectral geometric mean.

Convergence of Wasserstein mean to log-Euclidean mean under specific conditions.

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.