We introduce a method for jointly registering ensembles of partitioned datasets in a way which is both geometrically coherent and partition-aware. Once such a registration has been defined, one can group partition blocks across datasets in order to extract summary statistics, generalizing the commonly used order statistics for scalar-valued data. By modeling a partitioned dataset as an unordered $k$-tuple of points in a Wasserstein space, we are able to draw from techniques in optimal transport. More generally, our method is developed using the formalism of local Fr\'{e}chet means in symmetric products of metric spaces. We establish basic theory in this general setting, including Alexandrov curvature bounds and a verifiable characterization of local means. Our method is demonstrated on ensembles of political redistricting plans to extract and visualize basic properties of the space of plans for a particular state, using North Carolina as our main example.

中文翻译：

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分区数据集的几何平均值

我们介绍了一种以几何相干和分区感知的方式联合注册分区数据集集合的方法。一旦定义了这样的注册，就可以跨数据集对分区块进行分组，以提取汇总统计信息，概括标量值数据的常用顺序统计信息。通过将分区数据集建模为 Wasserstein 空间中点的无序 $k$-元组，我们能够利用最佳传输技术。更一般地说，我们的方法是使用度量空间的对称乘积中的局部 Fr\'{e}chet 均值的形式主义开发的。我们在这个一般设置中建立了基本理论，包括亚历山德罗夫曲率界限和局部均值的可验证特征。