BEGIN:VCALENDAR
PRODID:-//eluceo/ical//2.0/EN
VERSION:2.0
CALSCALE:GREGORIAN
BEGIN:VEVENT
UID:www.tcs.tifr.res.in/event/915
DTSTAMP:20230914T125943Z
SUMMARY:Betti Numbers of Gaussian Excursions in the Sparse Regime
DESCRIPTION:Speaker: Gugan Thoppe (Duke University\nDurham\, USA)\n\nAbstra
ct: \nAbstract: A function's excursion set is the sub-domain where its val
ue exceeds some threshold. Some key examples illustrating the central role
that excursion sets play in different application areas are as follows. I
n medical imaging\, in order to understand the brain parts involved in a p
articular task\, analysts frequently look at the high blood flow level reg
ions in the brain when the said task is being performed. In control theory
\, it is known that the viability and invariance properties of control sys
tems can be expressed as super-level sets of suitable value functions. In
robotics\, in order to plan its motion\, a sensor robot may want to identi
fy the sub-terrain where the accessibility probability is above some thres
hold. Often\, functions whose excursions are of interest are either random
themselves (for e.g.\, due to noise) or\, while being deterministic\, are
too complicated and hence can be treated as being a sample of a random fi
eld. In this sense\, studying the topology of random field excursions is v
ital. This work is the first detailed study of their Betti numbers (number
of holes) in the so-called `sparse' regime. Specifically\, we consider a
piecewise constant Gaussian field whose covariance function is positive an
d 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-v
anishing regime. We further have a Poisson approximation and a central lim
it theorem close to the transition threshold. Our proofs combine extreme v
alue theory and combinatorial topology tools (joint work with Sunder Ram K
rishnan).\n
URL:https://www.tcs.tifr.res.in/web/events/915
DTSTART;TZID=Asia/Kolkata:20181115T160000
DTEND;TZID=Asia/Kolkata:20181115T170000
LOCATION:A-201 (STCS Seminar Room)
END:VEVENT
END:VCALENDAR