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Gromov-Monge Quasimetrics and Distance Distributions

Banff International Research Station for Mathematical Innovation and Discovery

In applications in computer graphics and computational anatomy, one seeks a measure-preserving map from one shape to another which preserves geometry as much as possible. Inspired by this, we consider a notion of distance between arbitrary compact metric measure spaces by blending the Monge formulation of optimal transport with the Gromov-Hausdorff construction. We show that the resulting distance is an extended quasi-metric on the space of compact mm-spaces. This distance has convenient lower bounds defined in terms of distance distributions; these are functions associated to mm-spaces which have been used frequently as summaries in data and shape analysis applications. We provide rigorous results on the effectiveness of these lower bounds when restricted to simple classes of mm-spaces such as metric graphs or plane curves.This is joint work with Facundo Mémoli.

Mathematics; Calculus of variations and optimal control; optimization; Differential geometry

video/mp4

Author affiliation: Ohio State University

2019-06-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70611

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/