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Session S04 - Random Walks and Related Topics

Monday, July 19, 18:00 ~ 18:30 UTC-3

From First Passage Percolation to Topology Learning: Theory and Methods.

Pablo Groisman

Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

First Passage Percolation (FPP) has been studied for more than fifty years since the seminal work of Hammersley and Welsh in which they introduce it as a model of fluid flow through a porous (random) medium. Typically the model is known in advance and one wants to understand its behavior. Here we reverse the roles and use FPP to understand data. We know the behavior and want to understand the model (data). We will show that we can use FPP models to learn geometry, topology and chaotic dynamical systems.

More precisely, for N iid points in a manifold we introduce the Fermat distance which is nothing but the one introduced by Howard and Newman to consider Euclidean models of FPP and we study the set of points equipped with this distance as a metric space. We prove the convergence in the Gromov-Hausdorff sense to a metric space determined by the manifold and the density that produced the points. Then we use this result to show that we can use the Fermat distance to learn the topology of the manifold through persistent homology and to validate chaotic dynamical systems models .

Joint work with Facundo Sapienza (Berkley, USA), Matthieu Jonckheere (Universidad de Buenos Aires, Argentina), Ximena Fernández (Swansea University, UK), Eugenio Borghini (Universidad de Buenos Aires, Argentina) and Gabriel Mindlin (Universidad de Buenos Aires, Argentina).

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