2,271 research outputs found
Geometric influences
We present a new definition of influences in product spaces of continuous
distributions. Our definition is geometric, and for monotone sets it is
identical with the measure of the boundary with respect to uniform enlargement.
We prove analogs of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum
bounds for the new definition. We further prove an analog of a result of
Friedgut showing that sets with small "influence sum" are essentially
determined by a small number of coordinates. In particular, we establish the
following tight analog of the KKL bound: for any set in of
Gaussian measure , there exists a coordinate such that the th
geometric influence of the set is at least , where
is a universal constant. This result is then used to obtain an isoperimetric
inequality for the Gaussian measure on and the class of sets
invariant under transitive permutation group of the coordinates.Comment: Published in at http://dx.doi.org/10.1214/11-AOP643 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Note on the Entropy/Influence Conjecture
The entropy/influence conjecture, raised by Friedgut and Kalai in 1996, seeks
to relate two different measures of concentration of the Fourier coefficients
of a Boolean function. Roughly saying, it claims that if the Fourier spectrum
is "smeared out", then the Fourier coefficients are concentrated on "high"
levels. In this note we generalize the conjecture to biased product measures on
the discrete cube, and prove a variant of the conjecture for functions with an
extremely low Fourier weight on the "high" levels.Comment: 12 page
Geometric Influences II: Correlation Inequalities and Noise Sensitivity
In a recent paper, we presented a new definition of influences in product
spaces of continuous distributions, and showed that analogues of the most
fundamental results on discrete influences, such as the KKL theorem, hold for
the new definition in Gaussian space. In this paper we prove Gaussian analogues
of two of the central applications of influences: Talagrand's lower bound on
the correlation of increasing subsets of the discrete cube, and the
Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the
Gaussian results to obtain analogues of Talagrand's bound for all discrete
probability spaces and to reestablish analogues of the BKS theorem for biased
two-point product spaces.Comment: 20 page
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