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

In this paper we define a new weighted inductive geometric mean HnHn 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∈(0,1], the weighted geometric mean equation X=Hn+1(ωt;A1,…,An,X)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+t→0+ to the ω  -weighted Cartan mean of A1,…,AnA1,…,An. Numerical computations for the stochastic convergence in terms of HnHn to derive a new contractive barycenter other than the Cartan barycenter are presented.