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.
翻译:我们采用一种方法,共同登记被分割的数据集的集合,其方式既具有几何一致性,又具有分区意识。一旦界定了这种登记,就可以将数据集的分隔区块分组起来,以便提取简要统计数据,对通用的天平值数据顺序统计进行概括化。通过将分离数据集建模为瓦塞尔斯坦空间未经排序的点数图,我们可以从最佳运输技术中提取数据。更一般地说,我们的方法是利用本地Fr\{{e}chitch 手段的形式主义来开发的。我们在这个总体设置中建立了基本理论,包括亚历山德拉多夫曲线界限和可核实的当地手段特征。我们的方法体现在政治重新划分计划组合上,以北卡罗来纳州为主要例子,为特定州提取和直观计划空间的基本属性。