矩阵形式的海林格距离
Matrix versions of the Hellinger distance
摘要：
在正定矩阵空间上，我们考虑$d(A,B)=\left[\tr\mathcal{A}(A,B)-\tr\mathcal{G}(A,B)\right]^{1/2},$形式的距离函数，其中$\mathcal{A}(A,B)$是算术平均值，$\mathcal{G}(A,B)$是几何平均值的不同形式之一。当$\mathcal{G}(A,B)=A^{1/2}B^{1/2}$时，该距离为$\|A^{1/2}-B^{1/2}\|_2,$，当$\mathcal{G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$时，该距离为Bures-Wasserstein度量。我们研究了另外两种情况：$\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$的Pusz-Woronowicz几何平均值，和$\mathcal{G}(A,B)=\exp\big(\frac{\log A+\log B}{2}\big),$的对数欧几里德平均值。有了这些选择，$d(A,B)$不再是一个度量，但事实证明，$d^2(A,B)$是一个发散。我们建立了这些分歧的一些（严格）凸性性质。我们得到了$m$正定矩阵重心关于这些距离测度的特征。
英文摘要：
On the space of positive definite matrices we consider distance functions of the form $d(A,B)=\left[\tr\mathcal{A}(A,B)-\tr\mathcal{G}(A,B)\right]^{1/2},$ where $\mathcal{A}(A,B)$ is the arithmetic mean and $\mathcal{G}(A,B)$ is one of the different versions of the geometric mean. When $\mathcal{G}(A,B)=A^{1/2}B^{1/2}$ this distance is $\|A^{1/2}-B^{1/2}\|_2,$ and when $\mathcal{G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$ it is the Bures-Wasserstein metric. We study two other cases: $\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$ the Pusz-Woronowicz geometric mean, and $\mathcal{G}(A,B)=\exp\big(\frac{\log A+\log B}{2}\big),$ the log Euclidean mean. With these choices $d(A,B)$ is no longer a metric, but it turns out that $d^2(A,B)$ is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of $m$ positive definite matrices with respect to these distance measures.
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