Distance distributions and inverse problems for metric measure spaces

TL;DR

This paper develops a formal framework to evaluate how well distance distributions distinguish between different metric measure spaces, addressing their discriminative power and inverse problems across various categories.

Contribution

It introduces a rigorous approach to assess the discriminative ability of distance distributions and solves inverse problems in multiple categories of metric measure spaces.

Findings

Counterexample to the Curve Histogram Conjecture

Sphere rigidity results for Riemannian manifolds

Local injectivity in metric graphs

Abstract

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.