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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:453

Article

Hiai, Fumio¹  Lim, Yongdo² 

[1] Tohoku Univ, Hakusan 3-8-16-303, Abiko, Chiba 2701154, Japan

[2] Sungkyunkwan Univ, Dept Math, Suwon 440746, South Korea

关键词: Positive definite matrix;    Cartan barycenter;    Wasserstein distance;    Log-majorization;    Lie-Trotter formula;    Unitarily invariant norm;   

DOI  :  10.1016/j.jmaa.2017.03.027

来源: Elsevier

【 摘 要 】

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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free