Tensor product and Hadamard product for the Wasserstein means

TL;DR

This paper investigates properties of the Wasserstein mean of positive definite matrices, focusing on inequalities involving tensor and Hadamard products, and explores its behavior under certain linear maps.

Contribution

It establishes new inequalities for the Wasserstein mean related to tensor and Hadamard products, and characterizes its behavior under positive linear maps.

Findings

Verified inequalities of the Wasserstein mean with positive unital linear maps

Proved identity of the Wasserstein mean for tensor product

Derived inequalities for the Wasserstein mean with Hadamard product

Abstract

As one of the least squares mean, we consider the Wasserstein mean of positive definite Hermitian matrices. We verify in this paper the inequalities of the Wasserstein mean related with a strictly positive and unital linear map, the identity of the Wasserstein mean for tensor product, and some inequalities of the Wasserstein mean for Hadamard product.