TY - JOUR
T1 - A fixed point mean approximation to the Cartan barycenter of positive definite matrices
AU - Kim, Sejong
AU - Lee, Hosoo
AU - Lim, Yongdo
N1 - Publisher Copyright:
© 2016 Elsevier Inc. All rights reserved.
PY - 2016/5/1
Y1 - 2016/5/1
N2 - In this paper we define a new weighted inductive geometric mean Hn 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], the weighted geometric mean equation X=Hn+1(ωt;A1,...,An,X), where ωt=t1+t(w1,...,wn,1/t), has a unique positive definite solution that approaches as t→0+ to the ω-weighted Cartan mean of A1,...,An. Numerical computations for the stochastic convergence in terms of Hn to derive a new contractive barycenter other than the Cartan barycenter are presented.
AB - In this paper we define a new weighted inductive geometric mean Hn 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], the weighted geometric mean equation X=Hn+1(ωt;A1,...,An,X), where ωt=t1+t(w1,...,wn,1/t), has a unique positive definite solution that approaches as t→0+ to the ω-weighted Cartan mean of A1,...,An. Numerical computations for the stochastic convergence in terms of Hn to derive a new contractive barycenter other than the Cartan barycenter are presented.
KW - Cartan barycenter
KW - Fixed point geometric mean
KW - Multivariable geometric mean
KW - Positive definite matrix
KW - Riemannian trace metric
KW - Stochastic approximation
UR - https://www.scopus.com/pages/publications/84958020039
U2 - 10.1016/j.laa.2016.02.005
DO - 10.1016/j.laa.2016.02.005
M3 - Article
AN - SCOPUS:84958020039
SN - 0024-3795
VL - 496
SP - 420
EP - 437
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
ER -