We consider in this paper probability measures with compact support on the open convex cone of positive definite Hermitian matrices. We define the least squares barycenter for the Bures–Wasserstein distance, called the Wasserstein barycenter, as a unique minimizer of the integral of squared Bures–Wasserstein distances. Furthermore, interesting properties of the Wasserstein barycenter including the iteration approach via optimal transport maps, the boundedness and extended Lie–Trotter–Kato formula, and several inequalities with power means have been established in the setting of compactly supported measures.

中文翻译：

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紧支撑测度的 Wasserstein 重心

我们在本文中考虑了对正定 Hermitian 矩阵的开凸锥具有紧支持的概率测度。我们定义了 Bures-Wasserstein 距离的最小二乘重心，称为 Wasserstein 重心，作为平方 Bures-Wasserstein 距离积分的唯一最小值。此外，Wasserstein 重心的有趣特性包括通过最优传输图的迭代方法、有界性和扩展的 Lie-Trotter-Kato 公式，以及在紧支持测度的设置中建立的几个具有幂均值的不等式。