Fourier Entropy-Influence Conjecture for Random Linear Threshold Functions

  • Sourav Chakraborty
  • Sushrut Karmalkar
  • Srijita Kundu
  • Satyanarayana V. Lokam
  • Nitin Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function \(f:\{+1,-1\}^n \rightarrow \{+1,-1\}\), the Fourier entropy of f is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a “random” linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on \([-1,1]\) and Normal distribution.



We thank the reviewers for helpful comments that improved the presentation of the paper.


  1. 1.
    Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, Fourier transform, and learnability. J. ACM 40(3), 607–620 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boppana, R.B.: The average sensitivity of bounded-depth circuits. Inf. Process. Lett. 63(5), 257–261 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ganor, A., Komargodski, I., Lee, T., Raz, R.: On the noise stability of small Demorgan formulas, Technical report, Electronic Colloquium on Computational Complexity (ECCC) TR 12-174 (2012)Google Scholar
  4. 4.
    O’Donnell, R., Saks, M., Schramm, O.: Every decision tree has an influential variable. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005, pp. 31–39. IEEE Computer Society (2005)Google Scholar
  5. 5.
    O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, Cambridge (2014)CrossRefzbMATHGoogle Scholar
  6. 6.
    Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. Am. Math. Soc. 124(10), 2993–3002 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J., Kalai, G.: Influences of variables and threshold intervals under group symmetries. Geom. Funct. Anal. (GAFA) 7(3), 438–461 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mansour, Y.: An \(\cal{O}\)(n\(^{\log \log n}\)) learning algorithm for DNF under the uniform distribution. J. Comput. Syst. Sci. 50(3), 543–550 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gopalan, P., Kalai, A.T., Klivans, A.: Agnostically learning decision trees. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, STOC 2008, pp. 527–536 (2008)Google Scholar
  10. 10.
    Gopalan, P., Kalai, A., Klivans, A.R.: A query algorithm for agnostically learning DNF? In: 21st Annual Conference on Learning Theory - COLT 2008, 9–12 July 2008, Helsinki, Finland, pp. 515–516 (2008)Google Scholar
  11. 11.
    Friedgut, E.: Boolean functions with low average sensitivity depend on few coordinates. Combinatorica 18(1), 27–35 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kahn, J., Kalai, G., Linial, N.: The influence of variables on Boolean functions. In: Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, pp. 68–80 (1988)Google Scholar
  13. 13.
    O’Donnell, R., Wright, J., Zhou, Y.: The Fourier entropy–influence conjecture for certain classes of Boolean functions. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 330–341. Springer, Heidelberg (2011). CrossRefGoogle Scholar
  14. 14.
    Kalai, G.: The entropy/influence conjecture. Terence Tao’s blog:
  15. 15.
    Klivans, A., Lee, H., Wan, A.: Mansour’s conjecture is true for random DNF formulas. In: Proceedings of the 23rd Conference on Learning Theory, pp. 368–380 (2010)Google Scholar
  16. 16.
    Das, B., Pal, M., Visavaliya, V.: The entropy influence conjecture revisited. Technical report, arXiv:1110.4301 (2011)
  17. 17.
    O’Donnell, R., Tan, L.Y.: A composition theorem for the Fourier entropy-influence conjecture. In: Proceedings of Automata, Languages and Programming - 40th International Colloquium, pp. 780–791 (2013)Google Scholar
  18. 18.
    Chakraborty, S., Kulkarni, R., Lokam, S.V., Saurabh, N.: Upper bounds on Fourier entropy. Theor. Comput. Sci. 654, 92–112 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wan, A., Wright, J., Wu, C.: Decision trees, protocols and the entropy-influence conjecture. In: Innovations in Theoretical Computer Science, pp. 67–80 (2014)Google Scholar
  20. 20.
    Hicks, J.S., Wheeling, R.F.: An efficient method for generating uniformly distributed points on the surface of an n-dimensional sphere. Commun. ACM 2(4), 17–19 (1959)CrossRefzbMATHGoogle Scholar
  21. 21.
    Muller, M.E.: A note on a method for generating points uniformly on n-dimensional spheres. Commun. ACM 2(4), 19–20 (1959)CrossRefzbMATHGoogle Scholar
  22. 22.
    Marsaglia, G.: Choosing a point from the surface of a sphere. Ann. Math. Stat. 43(2), 645–646 (1972)CrossRefzbMATHGoogle Scholar
  23. 23.
    Petersen, W.P., Bernasconi, A.: Uniform sampling from an \(n\)-sphere. Technical report. Swiss Center for Scientific Computing (1997)Google Scholar
  24. 24.
    Szarek, S.J.: On the best constants in the Khinchine inequality. Studia Math. 58, 197–208 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shevtsova, I.: Moment-type estimates with asymptotically optimal structure for the accuracy of the normal approximation. Annales Mathematicae Et Informaticae 39, 241–307 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Boucheron, S., Lugosi, G., Massart, P.: Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford (2013)CrossRefzbMATHGoogle Scholar
  27. 27.
    David, H.A., Nagaraja, H.N.: Order Statistics, 3rd edn. Wiley, Hoboken (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, Hoboken (1968)zbMATHGoogle Scholar
  29. 29.
    Kane, D.M.: The correct exponent for the Gotsman-Linial Conjecture. Comput. Complex. 23(2), 151–175 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chennai Mathematical InstituteChennaiIndia
  2. 2.Centrum Wiskunde InformatikaAmsterdamNetherlands
  3. 3.University of TexasAustinUSA
  4. 4.Centre for Quantum TechnologiesSingaporeSingapore
  5. 5.Microsoft ResearchBangaloreIndia
  6. 6.Charles UniversityPragueCzech Republic

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