Combinatorica

, Volume 29, Issue 3, pp 363–387 | Cite as

On the Fourier spectrum of symmetric Boolean functions

  • Mihail N. Kolountzakis
  • Richard J. Lipton
  • Evangelos Markakis
  • Aranyak Mehta
  • Nisheeth K. Vishnoi
Article

Abstract

We study the following question

What is the smallest t such that every symmetric boolean function on κ variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t?

We exclude the constant functions for which there is no such t and the parity functions for which t has to be κ. Let τ (κ) be the smallest such t. Our main result is that for large κ, τ (κ)≤4κ/logκ.

The motivation for our work is to understand the complexity of learning symmetric juntas. A κ-junta is a boolean function of n variables that depends only on an unknown subset of κ variables. A symmetric κ-junta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric κ-juntas, in the uniform PAC learning model, in time n o(κ) . This improves on a result of Mossel, O’Donnell and Servedio in [16], who show that symmetric κ-juntas can be learned in time n 2κ/3.

Mathematics Subject Classification (2000)

42B05 68Q32 

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References

  1. [1]
    N. Alon, A. Andoni, T. Kaufman, K. Matulef, R. Rubinfeld and N. Xie: Testing κ-wise and almost κ-wise independence, in: STOC, pages 496–505, 2007.Google Scholar
  2. [2]
    A. Bernasconi: Mathematical Techniques for the Analysis of Boolean Functions, PhD thesis, Università degli Studi di Pisa, Dipartimento de Informatica, 1998.Google Scholar
  3. [3]
    A. Blum: Relevant examples and relevant features: Thoughts from computational learning theory; in: AAAI Symposium on Relevance, 1994.Google Scholar
  4. [4]
    A. Blum: Open problems, COLT, 2003.Google Scholar
  5. [5]
    A. Blum, M. Furst, M. Kearns and R. J. Lipton: Cryptographic primitives based on hard learning problems, in: CRYPTO, pages 278–291, 1993.Google Scholar
  6. [6]
    A. Blum and P. Langley: Selection of relevant features and examples in machine learning, Artificial Intelligence 97 (1997), 245–271.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    N. Bshouty, J. Jackson and C. Tamon: More efficient PAC learning of DNF with membership queries under the uniform distribution, in: Annual Conference on Computational Learning Theory, pages 286–295, 1999.Google Scholar
  8. [8]
    P. Cameron: Combinatorics: topics, techniques, algorithms; Cambridge University Press, 1994.Google Scholar
  9. [9]
    D. Helmbold, R. Sloan and M. Warmuth: Learning integer lattices, SIAM Journal of Computing 21(2) (1992), 240–266.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. Jackson: An efficient membership-query algorithm for learning dnf with respect to the uniform distribution, Journal of Computer and System Sciences 55 (1997), 414–440.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Kolountzakis, E. Markakis and A. Mehta: Learning symmetric juntas in time n o(κ), in: Proceedings of the conference Interface entre l’analyse harmonique et la theorie des nombres, CIRM, Luminy, 2005.Google Scholar
  12. [12]
    A. Kumchev: The distribution of prime numbers, manuscript, 2005.Google Scholar
  13. [13]
    N. Linial, Y. Mansour and N. Nisan: Constant depth circuits, fourier transform and learnability; Journal of the ACM 40(3) (1993), 607–620.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    R. Lipton, E. Markakis, A. Mehta and N. Vishnoi: On the fourier spectrum of symmetric boolean functions with applications to learning symmetric juntas, in: IEEE Conference on Computational Complexity (CCC), pages 112–119, 2005.Google Scholar
  15. [15]
    Y. Mansour: An o(n loglogn) learning algorithm for DNF under the uniform distribution, Journal of Computer and System Sciences 50 (1995), 543–550.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    E. Mossel, R. O’Donnell and R. Servedio: Learning juntas, in: STOC, pages 206-212, 2003.Google Scholar
  17. [17]
    G. Pólya and G. Szegő: Problems and theorems in Analysis, II; Springer, 1976.Google Scholar
  18. [18]
    T. Siegenthaler: Correlation-immunity of nonlinear combining functions for cryptographic applications, IEEE Transactions on Information Theory 30(5) (1984), 776–780.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    L. Valiant: A theory of the learnable, Communications of the ACM 27(11) (1984), 1134–1142.MATHCrossRefGoogle Scholar
  20. [20]
    K. Verbeurgt: Learning DNF under the uniform distribution in quasi-polynomial time, in: Annual Workshop on Computational Learning Theory, pages 314–326, 1990.Google Scholar
  21. [21]
    K. Verbeurgt: Learning sub-classes of monotone DNF on the uniform distribution, in: Algorithmic Learning Theory, 9th International Conference (Michael M. Richter, Carl H. Smith, Rolf Wiehagen, and Thomas Zeugmann, editors), pages 385–399, 1998.Google Scholar
  22. [22]
    J. von zur Gathen and J. Roche: Polynomials with two values, Combinatorica 17(3) (1997), 345–362.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2009

Authors and Affiliations

  • Mihail N. Kolountzakis
    • 1
  • Richard J. Lipton
    • 2
    • 3
  • Evangelos Markakis
    • 4
  • Aranyak Mehta
    • 5
  • Nisheeth K. Vishnoi
    • 6
    • 7
  1. 1.Department of MathematicsUniv. of CreteIraklioGreece
  2. 2.Georgia TechCollege of ComputingAtlantaUSA
  3. 3.Telcordia ResearchMorristownUSA
  4. 4.Centre for Math and Computer Science (CWI)AmsterdamThe Netherlands
  5. 5.IBM Almaden Research CenterSan JoseUSA
  6. 6.College of ComputingGeorgia Institute of TechnologyAtlantaUSA
  7. 7.IBM India Research LabNew DelhiIndia

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