Israel Journal of Mathematics

, Volume 225, Issue 1, pp 71–109 | Cite as

Gaussian noise sensitivity and Fourier tails

  • Guy Kindler
  • Naomi Kirshner
  • Ryan O’Donnell


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 ρ = cos( π/2ℓ), 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 \(||{f^{ > k}}||_2^2 \geqslant \Omega (Var[f]).\frac{1}{{\sqrt k }}for:{R^n} \to \{ - 1,1\} \). Combining this with the Invariance Principle shows the same Fourier tail bound for any Boolean f: {−1, 1} n → {−1, 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 \(\frac{1}{{\sqrt k {{\log }^{1.5}}k}}\) .


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [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), 111–122.CrossRefzbMATHGoogle Scholar
  2. [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), 4816–4819.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BH97]
    S. Bobkov and C. Houdré, Some Connections between Isoperimetric and Sobolevtype Inequalities, Memoirs of the American Mathematical Society 129 (1997).Google Scholar
  4. [Bor75]
    C. Borell, The Brunn–Minkowski inequality in Gauss space, Inventiones Mathematicae 30 (1975), 207–216.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Bor84]
    C. Borell, On polynomial chaos and integrability, Probabability and Mathematical Statistics 3 (1984), 191–203.MathSciNetzbMATHGoogle Scholar
  6. [Bor85]
    C. Borell, Geometric bounds on the Ornstein–Uhlenbeck velocity process, Probability Theory and Related Fields 70 (1985), 1–13.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Bou02]
    J. Bourgain, On the distribution of the Fourier spectrum of Boolean functions, Israel Journal of Mathematics 131 (2002), 269–276.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [CL90]
    E. Carlen and M. Loss, Extremals of functionals with competing symmetries, Journal of Functional Analysis 88 (1990), 437–456.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [GW95]
    M. Goemans and D. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the ACM 42 (1995), 1115–1145.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Kal02]
    G. Kalai, A Fourier-theoretic perspective on the Condorcet paradox and Arrow’s theorem, Advances in Applied Mathematics 29 (2002), 412–426.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Kan11a]
    D. Kane, The Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions, Computational Complexity 20 (2011), 389–412.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Kan11b]
    D. Kane, On Elliptic Curves, the ABC Conjecture, and Polynomial Threshold Functions, PhD thesis, Harvard University, 2011.Google Scholar
  13. [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.zbMATHGoogle Scholar
  14. [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), 319–357.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [KN06]
    S. Khot and A. Naor, Nonembeddability theorems via Fourier analysis, Mathematische Annalen 334 (2006), 821–852.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [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
  17. [KS88]
    W. Krakowiak and J. Szulga, Hypercontraction principle and random multilinear forms, Probability Theory and Related Fields 77 (1988), 325–342.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [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.Google Scholar
  19. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Led06]
    M. Ledoux, Personal communication, 2006.Google Scholar
  21. [MOO10]
    E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality Annals of Mathematics 171 (2010), 295–341.Google Scholar
  22. [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), 125–145.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [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.CrossRefzbMATHGoogle Scholar
  24. [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.CrossRefzbMATHGoogle Scholar
  25. [Tal96]
    M. Talagrand, How much are increasing sets positively correlated?, Combinatorica 16 (1996), 243–258.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations