# Parameterized Wasserstein mean with its properties

## TL;DR

This paper introduces a parameterized Wasserstein mean for positive definite matrices, exploring its properties, inequalities, and relations to other means, extending classical results like the Lie-Trotter-Kato formula.

## Contribution

It proposes a new mean generalizing the Wasserstein mean, analyzes its properties, bounds, and majorization relations, extending existing mathematical frameworks.

## Key findings

- Established norm inequalities and bounds for the mean.
- Extended the Lie-Trotter-Kato formula to this new mean.
- Proved log-majorization properties using the Cartan mean.

## Abstract

A new least squares mean of positive definite matrices for the divergence associated with the sandwiched quasi-relative entropy has been introduced. It generalizes the well-known Wasserstein mean for covariance matrices of Gaussian distributions with mean zero, so we call it the parameterized Wasserstein mean. We investigate in this article norm inequality of the parameterized Wasserstein mean, give its bounds with respect to the Loewner order, and show the extended version of Lie-Trotter-Kato formula for the parameterized Wasserstein mean. Finally we show the log-majorzation properties of the parameterized Wasserstein mean by using the Cartan mean.