  Applications in data science, shape analysis and object classification
frequently require comparison of probability distributions defined on different
ambient spaces. To accomplish this, one requires a notion of distance on a
given class of metric measure spaces -- that is, compact metric spaces endowed
with probability measures. Such distances are typically defined as comparisons
between metric measure space invariants, such as distance distributions (also
referred to as shape distributions, distance histograms or shape contexts in
the literature). Generally, distances defined in terms of distance
distributions are actually pseudometrics, in that they may vanish when
comparing nonisomorphic spaces. The goal of this paper is to set up a formal
framework for assessing the discrimininative power of distance distributions,
i.e., the extend to which these pseudometrics fail to define proper metrics. We
formulate several precise inverse problems in terms of these invariants and
answer them in several categories of metric measure spaces, including the
category of plane curves, where we give a counterexample to the Curve Histogram
Conjecture of Brinkman and Olver, the categories of embedded and Riemannian
manifolds, where we obtain sphere rigidity results, and the category of metric
graphs, where we obtain a local injectivity result along the lines of classical
work of Boutin and Kemper on point cloud configurations. The inverse problems
are further contextualized by the introduction of a variant of the
Gromov-Wasserstein distance on the space of metric measure spaces, which is
inspired by the original Monge formulation of optimal transport.
