# 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*: *R*^{n} → {−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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## References

- [AFR12]L. Ambrosio, A. Figalli and E. Runa,
*On sets of finite perimeter inWiener spaces: reduced boundary and convergence to halfspaces*, Atti della Accademia Nazionale dei Lincei. Rendictoni Lincei. Matematica e Applicazioni**24**(2013), 111–122CrossRefMATHGoogle Scholar - [Bec92]W. Beckner,
*Sobolev inequalities, the Poisson semigroup, and analysis on the sphere sn*, Proceedings of the National Academy of Sciences of the United States of America**89**(1992), 4816–4819MathSciNetCrossRefMATHGoogle Scholar - [BH97]S. Bobkov and C. Houdré,
*Some Connections between Isoperimetric and Sobolevtype Inequalities*, Memoirs of the American Mathematical Society**129**(1997).Google Scholar - [Bor75]C. Borell,
*The Brunn–Minkowski inequality in Gauss space*, Inventiones Mathematicae**30**(1975), 207–216MathSciNetCrossRefMATHGoogle Scholar - [Bor84]C. Borell,
*On polynomial chaos and integrability*, Probabability and Mathematical Statistics**3**(1984), 191–203MathSciNetMATHGoogle Scholar - [Bor85]C. Borell,
*Geometric bounds on the Ornstein–Uhlenbeck velocity process*, Probability Theory and Related Fields**70**(1985), 1–13MathSciNetCrossRefMATHGoogle Scholar - [Bou02]J. Bourgain,
*On the distribution of the Fourier spectrum of Boolean functions*, Israel Journal of Mathematics**131**(2002), 269–276MathSciNetCrossRefMATHGoogle Scholar - [CL90]E. Carlen and M. Loss,
*Extremals of functionals with competing symmetries*, Journal of Functional Analysis**88**(1990), 437–456MathSciNetCrossRefMATHGoogle Scholar - [GW95]M. Goemans and D. Williamson,
*Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming*, Journal of the ACM**42**(1995), 1115–1145MathSciNetCrossRefMATHGoogle Scholar - [Kal02]G. Kalai,
*A Fourier-theoretic perspective on the Condorcet paradox and Arrow’s theorem*, Advances in Applied Mathematics**29**(2002), 412–426MathSciNetCrossRefMATHGoogle Scholar - [Kan11a]D. Kane,
*The Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions*, Computational Complexity**20**(2011), 389–412MathSciNetCrossRefMATHGoogle Scholar - [Kan11b]D. Kane,
*On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions*, PhD thesis, Harvard University, 2011.Google Scholar - [Kho02]S. Khot,
*On the power of unique 2-prover 1-round games*, in*Proceedings of the 34th Annual ACM Symposium on Theory of Computing*, ACM, New York, 2002, pp. 767–775.Google Scholar - [KKMO07]S. Khot, G. Kindler, E. Mossel and R. O’Donnell,
*Optimal inapproximability results for Max-Cut and other 2-variable CSPs?*, SIAM Journal on Computing**37**(2007), 319–357MathSciNetCrossRefMATHGoogle Scholar - [KN06]S. Khot and A. Naor,
*Nonembeddability theorems via Fourier analysis*, Mathematische Annalen**334**(2006), 821–852MathSciNetCrossRefMATHGoogle Scholar - [KOS08]A. Klivans, R. O’Donnell and R. Servedio,
*Learning geometric concepts via Gaussian surface area*, in*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science*, IEEE Computer Society, Los Alamitos, CA, 2008, pp. 541–550.Google Scholar - [KS88]W. Krakowiak and J. Szulga,
*Hypercontraction principle and random multilinear forms*, Probability Theory and Related Fields**77**(1988), 325–342MathSciNetCrossRefMATHGoogle Scholar - [KV05]S. Khot and N. Vishnoi,
*The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1*, Journal of the ACM 62 (2015), Art. 8.MATHGoogle Scholar - [Led96]M. Ledoux,
*Isoperimetry and Gaussian analysis*, in*Lectures on Probability Theory and Statistics (Saint-Flour 1994), Lecture Notes in Mathematics*, Vol. 1648, Springer, Berlin, 1996, pp. 165–294.MathSciNetCrossRefMATHGoogle Scholar - [Led06]M. Ledoux,
*Personal communication*, 2006.Google Scholar - [MOO10]E. Mossel, R. O’Donnell and K. Oleszkiewicz,
*Noise stability of functions with low influences: invariance and optimality Annals of Mathematics***171**(2010), 295–341Google Scholar - [MPPP07]M. Miranda, D. Pallara, F. Paronetto and M. Preunkert,
*Short-time heat flow and functions of bounded variation in RN*, Annales de la Faculté des Sciences de Toulouse. Mathématiques**16**(2007), 125–145MathSciNetCrossRefMATHGoogle Scholar - [She99]W. Sheppard,
*On the application of the theory of error to cases of normal distribution and normal correlation*, Philosophical Transactions of the Royal Society of London. Series A**192**(1899), 101–167, 531.CrossRefMATHGoogle Scholar - [ST78]V. Sudakov and B. Tsirel’son,
*Extremal properties of half-spaces for spherically invariant measures*, Journal of Soviet Mathematics 9 (1978), 9–18; originally published in Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im A. M. Steklova**41**(1974), 14–21, 165.MathSciNetGoogle Scholar - [Tal96]M. Talagrand,
*How much are increasing sets positively correlated?*, Combinatorica**16**(1996), 243–258MathSciNetCrossRefMATHGoogle Scholar