Abstract

We propose new tools for the geometric exploration of data objects taking values in a general separable metric space. For a random object, we first introduce the concept of distance profiles. Specifically, the distance profile of a point in a metric space is the distribution of distances between the point and the random object. Distance profiles can be harnessed to define transport ranks based on optimal transport, which capture the centrality and outlyingness of each element in the metric space with respect to the probability measure induced by the random object. 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. In particular, we establish the theoretical guarantees for the estimation of the distance profiles and the transport ranks for a wide class of metric spaces, followed by practical illustrations.