Log-majorization and Lie-Trotter formula for the Cartan barycenter on probability measure spaces

TL;DR

This paper generalizes log-majorization and the Lie-Trotter formula for the Cartan barycenter from matrices to probability measures on positive definite matrices, establishing new inequalities and monotonicity properties.

Contribution

It extends log-majorization to the Cartan barycenter for probability measures and proves a Lie-Trotter formula in this broader setting.

Findings

Established log-majorization for the Cartan barycenter of probability measures.

Derived a Lie-Trotter formula for the Cartan barycenter.

Proved monotonicity of the Cartan barycenter map under stochastic order.

Abstract

We extend Ando-Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie-Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the main application of log-majorization.