Facundo Mémoli – Rutgers University
Date/Time/Location
Thursday, February 12, 2026, 12:10 pm; Hill Center 705
Distance distributions and inverse problems for metric measure spaces
Many problems in data science—shape analysis, object recognition, and object classification—require comparing probability distributions that live in different ambient spaces. A convenient setting for this is the class of metric measure spaces, i.e., compact metric spaces equipped with probability measures. A widely used approach compares such spaces via computable invariants such as distance distributions (also called shape distributions, distance histograms, or shape contexts). Distances built from these invariants are typically only pseudometrics: they can assign distance zero to non-isomorphic spaces.
In this talk, I will address the basic question: how much information is contained in a distance distribution? I will formulate precise inverse problems asking when distance distributions determine the underlying metric measure space, and I will present answers in several settings. These include 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 rigidity results for spheres; and the category of metric graphs, where we prove a local injectivity result in the spirit of classical work of Boutin and Kemper on point configurations. These results clarify the reach and limitations of distance-distribution–based methods for comparing shapes and geometric data.