A fixed point mean approximation to the Cartan barycenter of positive definite matrices

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• 作者：Sejong Kim^a ; ^{skim@chungbuk.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Hosoo Lee^b ; ^{hosoo@yu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Yongdo Lim^c ; ^{ylim@skku.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 47A64 ; secondary ; 47A63 ; 47L25
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：496
• 期：Complete
• 页码：420-437
• 全文大小：357 K

文摘

In this paper we define a new weighted inductive geometric mean H_n$H_n$ on the Riemannian manifold of positive definite matrices of fixed size and present its fixed point mean approximation to the Cartan barycenter (Karcher mean). We show that for t∈(0,1]$t\in (0,1]$, the weighted geometric mean equation X=H_n+1(ω_t;A₁,…,A_n,X)$X=H_{n+1}({\omega }_t;A_1,\dots ,A_n,X)$, where ${\omega }_t=\frac{t}{1+t}(w_1,\dots ,w_n,1/t)$, has a unique positive definite solution that approaches as t→0⁺$t\to 0^{+}$ to the ω  -weighted Cartan mean of A₁,…,A_n$A_1,\dots ,A_n$. Numerical computations for the stochastic convergence in terms of H_n$H_n$ to derive a new contractive barycenter other than the Cartan barycenter are presented.