Distributions of distances and volumes of balls in homogeneous lens spaces

Balch, Brenden; Peterson, Chris; Shonkwiler, Clayton

doi:10.1016/j.difgeo.2020.101712

Cited by 1 publication

(2 citation statements)

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“…">3.For compact metric graphs X and Y with uniform measures, does the existence of a map

\phi\! :X \rightarrow Y

such that

dh_\mathcal {X} = dh_\mathcal {Y} \circ \phi

imply X and Y are isomorphic? Questions regarding the use of volume‐growth information to characterize a space have been studied frequently in the Riemannian setting, 8,17–23 where infinitesimal volume growth is related to scalar curvature, as well as for certain mm‐spaces embedded in Euclidean space and endowed with extrinsic distance 24,25 . Inverse problems with a similar flavor have also been studied recently in the topological data analysis literature, where the goal is to characterize rougher spaces such as metric graphs via families of topological signatures 26–30 .…”

Section: Introductionmentioning

confidence: 99%

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Distance distributions and inverse problems for metric measure spaces

Mémoli¹,

Needham

2022

Stud Appl Math

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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, that is, the extent 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.

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“…">3.For compact metric graphs X and Y with uniform measures, does the existence of a map