Advertisement

Talagrand Inequality at Second Order and Application to Boolean Analysis

  • Kevin TanguyEmail author
Article
  • 6 Downloads

Abstract

This note is concerned with an extension, at second order, of an inequality on the discrete cube \(C_n=\{-\,1,1\}^n\) (equipped with the uniform measure) due to Talagrand (Ann Probab 22:1576–1587, 1994). As an application, the main result of this note is a theorem in the spirit of a famous result from Kahn et al. (cf. Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol 62. Computer Society Press, Washington, pp 68–80, 1988) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates \((i,j)\in \{1,\ldots ,n\}^2\) is introduced in Sect. 2, and the following alternative is obtained: For any Boolean function \(f\,:\, C_n\rightarrow \{0,1\}\), either there exists a coordinate with influence at least of order \((1/n)^{1/(1+\eta )}\), with \(\, 0<\eta <1\) (independent of f and n), or there exists a couple of coordinates \((i,j)\in \{1,\ldots ,n\}^2\) with \(i\ne j\), with influence at least of order \((\log n/n)^2\). In Sect. 4, it is shown that this extension of Talagrand inequality can also be obtained, with minor modifications, for the standard Gaussian measure \(\gamma _n\) on \({\mathbb {R}}^n\); the obtained inequality can be of independent interest. The arguments rely on interpolation methods by semigroup together with hypercontractive estimates. At the end of the article, some related open questions are presented.

Keywords

Boolean analysis Influences Hypercontractivity Functional inequalities 

Mathematics Subject Classification (2010)

60E15 60J25 06E30 28A12 

Notes

Acknowledgements

This work was initiated during my thesis and I thank my Ph.D. advisor M. Ledoux for introducing this problem to me and for fruitful discussions. I am also indebted to K. Oleszkiewicz for several comments and precious advice. I also want to thank C. Houdré for kindly pointing out to me the reference [13] and R. Kumolka for his linguistic help. Finally, I warmly thank the anonymous referee and R. Bouyrie for helpful comments in improving the exposition.

References

  1. 1.
    Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators. In: Grundlehren der Mathematischen Wissenschaften, vol. 348, (2014)Google Scholar
  2. 2.
    Ben-Or, M., Linial, N.: Collective Coin Flipping, p. 70. Randomness and Computation Academic Press, Cambridge (1990)Google Scholar
  3. 3.
    Bobkov, S., Götze, F., Houdré, C.: On Gaussian and Bernoulli covariance representations. Bernoulli 7(3), 439–451 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bobkov, S., Götze, F., Sambale, H.: Higher order concentration of measure. Preprint, arXiv:1709.06838, (2017)
  5. 5.
    Boucheron, T., Lugosi, G., Massart, P.: Concentration Inequalities: A Nonasymptotic Theory of Independance. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar
  6. 6.
    Bouyrie, R.: Private communication. (2017)Google Scholar
  7. 7.
    Chatterjee, S.: Superconcentration and Related Topics. Springer, Cham (2014)CrossRefGoogle Scholar
  8. 8.
    Cordero-Erausquin, D., Ledoux, M.: Hypercontractive measures, Talagrand’s inequality, and influences. In: Geometric Aspects of Functional Analysis, Lectures Notes in Mathematics, vol. 2050, pp. 169–189, (2012)Google Scholar
  9. 9.
    Diaconis, P., Saloff-Coste, L.: Logarithmic Sobolev inequalities for finite Markov chains. Ann. Probab. 6(3), 695–750 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. Am. Math. Soc. 124(10), 2993–3002 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Garban, Christophe, Steif, Jeffrey E.: Noise Sensitivity of Boolean Functions and Percolation. Institute of Mathematical Statistics Textbooks. Cambridge University Press, New York (2015)CrossRefGoogle Scholar
  12. 12.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Houdré, C., Perez-Abreu, V., Surgailis, D.: Interpolation, correlation identities, and inequalities for infinitely divisible variables. J. Fourier Anal. Appl. 4(6), 651–668 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kahn, J., Kalai, G., Linial, N.: The influence of variables on Boolean functions. In: Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol. 62, pp. 68–80. Computer Society Press, Washington, (1988)Google Scholar
  15. 15.
    Keller, N., Mossel, E.: Quantitative relationship between noise sensitivity and influences. Combinatorica 33, 45–71 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Keller, N., Mossel, E., Sen, A.: Geometric influences. Ann. Probab. 40(3), 1135–1166 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ledoux, M.: L’algèbre de Lie des gradients itérés d’un générateur markovien - développements de moyenne et entropies. Ann. Sci. Ec. Norm. Super 28(4), 435–460 (1995)CrossRefGoogle Scholar
  18. 18.
    Ledoux, M.: The concentration of measure phenomenon. In: Mathematical Surveys and Monographs, vol. 89, (2001)Google Scholar
  19. 19.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (2011)zbMATHGoogle Scholar
  20. 20.
    Nelson, E.: The free markov field. J. Funct. Anal. 12, 211–227 (1973)CrossRefGoogle Scholar
  21. 21.
    O’Donnell, R.: Analysis of Boolean functions. Cambridge University Press, New York (2014)CrossRefGoogle Scholar
  22. 22.
    Oleszkiewicz, K.: Private communication. (2018)Google Scholar
  23. 23.
    Paouris, G., Valettas, P.: Dichotomies, structure, and concentration results. Preprint, arXiv:1708.05149, (2017)
  24. 24.
    Paouris, G., Valettas, P.: A gaussian small deviation inequality for convex functions. Ann. Probab. 46(3), 1141–1454 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rossignol, R.: Private communicationGoogle Scholar
  26. 26.
    Talagrand, M.: On Russo’s approximate zero-one law. Ann. Probab. 22, 1576–1587 (1994)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tanguy, K.: Some superconcentration inequalities for extrema of stationary gaussian processes. Stat. Probab. Lett. 106, 239–246 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tanguy, K.: Quelques inégalités de superconcentration: théorie et applications (in french). Ph.D. Thesis, Institute of Mathematics of Toulouse, Toulouse (2017)Google Scholar
  29. 29.
    Tanguy, K.: Non asymptotic variance bounds and deviation inequalities by optimal transport. Electron. J. Probab. 24(12), 1–18 (2019)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LAREMA (CNRS UMR 6093)University of AngersAngersFrance

Personalised recommendations