Skip to main content
SpringerLink
Log in

The Fourier Entropy–Influence Conjecture for Certain Classes of Boolean Functions

  • Conference paper
Automata, Languages and Programming (ICALP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6755))

Included in the following conference series:

Abstract

In 1996, Friedgut and Kalai made the Fourier Entropy– Influence Conjecture: For every Boolean function \(f : \{-1, 1\}^{n} \longrightarrow \{-1, 1\}\) it holds that \(H[{\widehat{f}}^{2}] \leq C \cdot I[f]\), where \(H[{\widehat{f}}^{2}]\) is the spectral entropy of f, I[f] is the total influence of f, and C is a universal constant. In this work we verify the conjecture for symmetric functions. More generally, we verify it for functions with symmetry group \(S_{n_1} \times \cdots \times S_{n_d}\) where d is constant. We also verify the conjecture for functions computable by read-once decision trees.

This research performed while the first author was a member of the School of Mathematics, Institute for Advanced Study. Supported by NSF grants CCF-0747250 and CCF-0915893, BSF grant 2008477, and Sloan and Okawa fellowships.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ben-Or, M., Linial, N.: Collective coin flipping. In: Micali, S. (ed.) Randomness and Computation. Academic Press, New York (1990)

    Google Scholar 

  2. Bernstein, A.: Maximally connected arrays on the n-cube. SIAM Journal on Applied Mathematics, 1485–1489 (1967)

    Google Scholar 

  3. Boppana, R.: The average sensitivity of bounded-depth circuits. Information Processing Letters 63(5), 257–261 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. De, A., Etesami, O., Trevisan, L., Tulsiani, M.: Improved pseudorandom generators for depth 2 circuits. In: Proceedings of the 14th Annual International Workshop on Randomized Techniques in Computation, pp. 504–517 (2010)

    Google Scholar 

  5. Feller, W.: An introduction to probability theory and its applications, 3rd edn., vol. 1. Wiley, Chichester (1968)

    MATH  Google Scholar 

  6. Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proceedings of the American Mathematical Society 124(10), 2993–3002 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gopalan, P., Kalai, A., Klivans, A.: Agnostically learning decision trees. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 527–536 (2008)

    Google Scholar 

  8. Gopalan, P., Kalai, A., Klivans, A.: A query algorithm for agnostically learning DNF? In: Proceedings of the 21st Annual Conference on Learning Theory, pp. 515–516 (2008)

    Google Scholar 

  9. Gross, L.: Logarithmic Sobolev inequalities. American Journal of Mathematics 97(4), 1061–1083 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harper, L.: Optimal assignments of numbers to vertices. Journal of the Society for Industrial and Applied Mathematics 12(1), 131–135 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Kalai, G.: The entropy/influence conjecture. Posted on Terence Tao’s What’s new blog (2007), http://terrytao.wordpress.com/2007/08/16/gil-kalai-the-entropyinfluence-conjecture/

  13. Keilson, J., Gerber, H.: Some results for discrete unimodality. Journal of the American Statistical Association 66(334), 386–389 (1971)

    Article  MATH  Google Scholar 

  14. Klivans, A., Lee, H., Wan, A.: Mansours Conjecture is true for random DNF formulas. Electronic Colloquium on Computational Complexity TR10-023 (2010)

    Google Scholar 

  15. Mansour, Y.: Learning boolean functions via the Fourier Transform. In: Roychowdhury, V., Siu, K.-Y., Orlitsky, A. (eds.) Theoretical Advances in Neural Computation and Learning, ch. 11, pp. 391–424. Kluwer Academic Publishers, Dordrecht (1994)

    Chapter  Google Scholar 

  16. Mansour, Y.: An O(n loglogn) learning algorithm for DNF under the uniform distribution. Journal of Computer and System Sciences 50(3), 543–550 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. O’Donnell, R.: Some topics in analysis of boolean functions. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 569–578 (2008)

    Google Scholar 

  18. O’Donnell, R., Saks, M., Schramm, O., Servedio, R.: Every decision tree has an influential variable. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 31–39 (2005)

    Google Scholar 

  19. Steiglitz, K., Bernstein, A.: Optimal binary coding of ordered numbers. Journal of the Society for Industrial and Applied Mathematics 13(2), 441–443 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. Talagrand, M.: On Russo’s approximate zero-one law. Annals of Probability 22(3), 1576–1587 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Titsworth, R.: Correlation properties of cyclic sequences. PhD thesis, California Institute of Technology (1962)

    Google Scholar 

  22. Wan, A.: Talk at the Center for Computational Intractability (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, USA

    Ryan O’Donnell, John Wright & Yuan Zhou

Authors
  1. Ryan O’Donnell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Wright
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. ICE-TCS, School of Computer Science, Reykjavik University, Menntavegur 1, IS 101, Reykjavik, Iceland

    Luca Aceto

  2. Fakultät für Informatik, Universität Wien, Universitätsstraße10/9, 1090, Wien, Österreich, Austria

    Monika Henzinger

  3. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Sgall

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

O’Donnell, R., Wright, J., Zhou, Y. (2011). The Fourier Entropy–Influence Conjecture for Certain Classes of Boolean Functions. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22006-7_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22005-0

  • Online ISBN: 978-3-642-22006-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation