Geometry of Comparisons

Many data analysis problems can be cast as distance geometry problems in \emph{space forms} -- Euclidean, elliptic, or hyperbolic spaces. We ask: what can be said about the underlying space form if we are only given a subset of comparisons between pairwise distances, without computing an actual embedding? To study this question, we define the \textit{ordinal capacity} of a metric space. Ordinal capacity measures how well a space can accommodate a given set of ordinal measurements. We prove that the ordinal capacity of a space form is related to its dimension and curvature sign, and provide a lower bound on the embedding dimension of non-metric graphs in terms of the \textit{ordinal spread} of their sub-cliques. Finally, we show how the statistics of ordinal spread can be used to identify the underlying space form of similarity graphs on weighted trees and gene expression data.