Log-majorization and Lie-Trotter formula for the Cartan barycenter on probability measure spaces
Log-majorization and Lie-Trotter formula for the Cartan barycenter on probability measure spaces
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概率测度空间上嘉当重心的对数极大化和 Lie-Trotter 公式
DOI：
10.1016/j.jmaa.2017.03.027
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
F. Hiai and Y. Lim
中科院分区：
文献类型：
--
作者：
関口次郎;F. Hiai and Y. Lim
我们将 Ando-Hiai 的正定矩阵加权几何平均值的对数极大化扩展到配备有迹度量的正定矩阵黎曼流形的概率测度的一般设置中的嘉当重心。主要关键是解决锥体诱发的随机阶数有限初矩概率测度空间上嘉当重心映射的单调性问题。我们还推导了嘉当重心的 Lie-Trotter 公式和相关酉不变范数不等式，作为对数极大化的主要应用。
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.