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Metrics on the collection of dynamic shapes

Banff International Research Station for Mathematical Innovation and Discovery

When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of groups of animals. In a similar vein, when attempting to classify motion capture data according to action one is confronted with having to match/compare shapes that evolve with time. Motivated by these applications, we study the question of suitably metrizing the collection of all dynamic metric spaces (DMSs). We construct a suitable metric on this collection and prove the stability of several natural invariants of DMSs under this metric. In particular, we prove that certain zigzag persistent homology invariants related to dynamic clustering are stable w.r.t. this distance. These lower bounds permit the efficient classification of dynamic shape data in applications. We will show computational experiments on dynamic data generated via distributed behavioral models. This is joint work with Woojin Kim and Zane Smith https://research.math.osu.edu/networks/formigrams/

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

video/mp4

Author affiliation: The Ohio State University

2019-06-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70610

Unreviewed

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