In this paper, we focus on the analysis of the regularized Wasserstein barycenter problem. We provide uniqueness and a characterization of the barycenter for two important classes of probability measures: (i) Gaussian distributions and (ii) $q$-Gaussian distributions; each regularized by a particular entropy functional. We propose an algorithm based on gradient projection method in the space of matrices in order to compute these regularized barycenters. We also consider a general class of $\varphi$-exponential measures, for which only the non-regularized barycenter is studied. Finally, we numerically show the influence of parameters and stability of the algorithm under small perturbation of data.

翻译：在本文中,我们侧重于分析正规化的瓦森斯坦中枢问题。我们为两种重要的概率计量类别提供了独特的和对中枢的定性:(一) 高山分布和(二) 高山分配;以及(二) 美元-加西元分配;每种分配都由特定的酶功能规范化。我们提出了基于基质空间梯度预测法的算法,以计算这些正规化的中枢。我们还考虑了一种普通的 $\varphie$-Explential措施,只研究非正规化的中枢。最后,我们用数字显示数据小扰动下算法参数和稳定性的影响。