Wasserstein barycenters in the manifold of all positive definite matrices

In this paper, we study the Wasserstein barycenter of finitely many Borel probability measures on $\mathbb {P}_{n}$, the Riemannian manifold of all $n\times n$ real positive definite matrices as well as its associated dual problem, namely the optimal transport problem. Our results generalize some results of Agueh and Carlier on $\mathbb {R}^{n}$ to $\mathbb {P}_{n}$. We show the existence of the optimal solutions and the Wasserstein barycenter measure. Furthermore, via a discretization approach and using the BFGS (Broyden-Fletcher-Goldfarb-Shanno) method for nonsmooth convex optimization, we propose a numerical method for computing the potential functions of the optimal transport problem. Also, thanks to the so-called optimal transport Jacobian on Riemannian manifolds of Cordero-Erausquin, McCann, and Schmuckenschläger, we show that the density of the Wasserstein barycenter measure can be approximated numerically. The paper concludes with some numerical experiments.