We propose new tools for the geometric exploration of data objects taking values in a general separable metric space $(\Omega, d)$. Given a probability measure on $\Omega$, we introduce depth profiles, where the depth profile of an element $\omega\in\Omega$ refers to the distribution of the distances between $\omega$ and the other elements of $\Omega$. Depth profiles can be harnessed to define transport ranks, which capture the centrality of each element in $\Omega$ with respect to the entire data cloud based on optimal transport maps between depth profiles. We study the properties of transport ranks and show that they provide an effective device for detecting and visualizing patterns in samples of random objects and also entail notions of transport medians, modes, level sets and quantiles for data in general separable metric spaces. Specifically, we study estimates of depth profiles and transport ranks based on samples of random objects and establish the convergence of the empirical estimates to the population targets using empirical process theory. We demonstrate the usefulness of depth profiles and associated transport ranks and visualizations for distributional data through a sample of age-at-death distributions for various countries, for compositional data through energy usage for U.S. states and for network data through New York taxi trips.

翻译：我们提出针对取值于一般可分度量空间$(\Omega, d)$中数据对象几何探索的新工具。给定$\Omega$上的概率测度，我们引入深度剖面概念，其中元素$\omega\in\Omega$的深度剖面指$\omega$与$\Omega$中其他元素间距离的分布。深度剖面可用于定义输运秩，该秩基于深度剖面间的最优输运映射，刻画每个元素$\Omega$在整个数据云中的中心性。我们研究输运秩的性质，证明其可作为检测与可视化随机对象样本中模式的有效工具，并衍生出一般可分度量空间中数据的输运中位数、众数、水平集与分位数等概念。具体而言，我们基于随机对象样本研究深度剖面与输运秩的估计，并利用经验过程理论建立经验估计向总体目标的收敛性。通过各国死亡年龄分布数据（分布型数据）、美国各州能源使用数据（成分型数据）及纽约出租车行程数据（网络型数据），我们展示了深度剖面及其关联输运秩与可视化方法在分布型数据中的实用价值。