Let $\mathcal{P}_{2,ac}$ be the set of Borel probabilities on $\mathbb{R}^d$
with finite second moment and absolutely continuous with respect to Lebesgue
measure. We consider the problem of finding the barycenter (or Fr\'echet mean)
of a finite set of probabilities $\nu_1,\ldots,\nu_k \in \mathcal{P}_{2,ac}$
with respect to the $L_2-$Wasserstein metric. For this task we introduce an
operator on $\mathcal{P}_{2,ac}$ related to the optimal transport maps pushing
forward any $\mu \in \mathcal{P}_{2,ac}$ to $\nu_1,\ldots,\nu_k$. Under very
general conditions we prove that the barycenter must be a fixed point for this
operator and introduce an iterative procedure which consistently approximates
the barycenter. The procedure allows effective computation of barycenters in
any location-scatter family, including the Gaussian case. In such cases the
barycenter must belong to the family, thus it is characterized by its mean and
covariance matrix. While its mean is just the weighted mean of the means of the
probabilities, the covariance matrix is characterized in terms of their
covariance matrices $\Sigma_1,\dots,\Sigma_k$ through a nonlinear matrix
equation. The performance of the iterative procedure in this case is
illustrated through numerical simulations, which show fast convergence towards
the barycenter.

本文介绍了一个迭代方法来计算有限 Borel 概率集合的质心，并主要探讨了其中在线性（加权平均值）和非线性（协方差矩阵）方面的性质以及应用方法。