Gaussian Noise Sensitivity and Fourier Tails

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Abstract

We observe a subadditivity property for the noise sensitivity of subsets of Gaussian space. For subsets of volume 1/2, this leads to an almost trivial proof of Borell’s Isoperimetric Inequality for \(\rho = \cos (\frac{\pi }{{2\ell }})\), ℓ ∈ N Rotational sensitivity also easily gives the Gaussian Isoperimetric Inequality for volume-1/2 sets and a.8787-factor UG-hardness for Max-Cut (within 10−4 of the optimum). As another corollary we show the Hermite tail bound \(||f > k||\Omega (Var[f]).\frac{1}{{\sqrt k }}\) for f: Rn → {−1, }. Combining this with the Invariance Principle shows the same Fourier tail bound for any Boolean f: {−1, } n → {−1, } with all its noisy-influences small, or more strongly, that a Boolean function with tail weight smaller than this bound must be close to a junta. This improves on a result of Bourgain, where the bound on the tail weight was only \(\frac{1}{{{k^{1/2 + o(1)}}}}\). We also show a simplification of Bourgain’s proof that does not use Invariance and obtains the bound \(\frac{1}{{\sqrt k {{\log }^{1.5}}k}}\).

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.School of Computer Science and EngineeringThe Hebrew University of JerusalemGivat Ram, JerusalemIsrael
  2. 2.Department of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

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