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

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.