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Betti numbers of Gaussian excursions in the sparse regime

Publication Type : Journal Article

Publisher : ArXiv

Source : Aug. 2018

Url :

Campus : Amritapuri

School : School of Computing

Department : Computer Science and Engineering

Year : 2018

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 obtain a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools.

Cite this Research Publication : G. Thoppe and S. R. Krishnan∗, “Betti numbers of Gaussian excursions in the sparse regime”, Aug. 2018

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