Topology, Geometry and Data Seminar - Gugan Thoppe

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Gugan Thoppe
September 11, 2018
4:10PM - 5:05PM
Location
Cockins Hall 240

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Add to Calendar 2018-09-11 16:10:00 2018-09-11 17:05:00 Topology, Geometry and Data Seminar - Gugan Thoppe Title: Betti Numbers of Gaussian Excursions in the Sparse Regime Speaker: Gugan Thoppe (Duke University) Abstract: Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asymptotic mean and variance estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We further have a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools. This is joint work with Sunder Ram Krishnan. Seminar URL: https://tgda.osu.edu/activities/tdga-seminar/ Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public
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Title: Betti Numbers of Gaussian Excursions in the Sparse Regime

SpeakerGugan Thoppe (Duke University)

Abstract: Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asymptotic mean and variance estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We further have a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools.

This is joint work with Sunder Ram Krishnan.

Seminar URL: https://tgda.osu.edu/activities/tdga-seminar/

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