Near-order relation of power means

TL;DR

This paper investigates the near-order properties of various power means of positive definite operators, revealing monotonicity and convergence behaviors among spectral geometric, Wasserstein, and Re9nyi power means.

Contribution

It introduces new near-order relationships and monotonicity results for power means of positive definite operators, including convergence to the log-Euclidean mean.

Findings

Monotonicity of spectral geometric and Wasserstein means on parameters.

Near-order relationship between spectral geometric and Wasserstein means.

Convergence of Re9nyi power mean to the log-Euclidean mean.

Abstract

On the setting of positive definite operators we study the near-order properties of power means such as the quasi-arithmetic mean (H\"{o}lder mean) and R\'{e}nyi power mean. We see the monotonicity of spectral geometric mean and Wasserstein mean on parameters with respect to the near-order and the near-order relationship between the spectral geometric mean and Wasserstein mean. Furthermore, the monotonicity of quasi-arithmetic mean on parameters and the convergence of R\'{e}nyi power mean to the log-Euclidean mean with respect to the near-order have been established.