On the Fourier spectrum of symmetric Boolean functions

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.

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.

  2. [2]

    A. Bernasconi: Mathematical Techniques for the Analysis of Boolean Functions, PhD thesis, Università degli Studi di Pisa, Dipartimento de Informatica, 1998.

  3. [3]

    A. Blum: Relevant examples and relevant features: Thoughts from computational learning theory; in: AAAI Symposium on Relevance, 1994.

  4. [4]

    A. Blum: Open problems, COLT, 2003.

  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.

  6. [6]

    A. Blum and P. Langley: Selection of relevant features and examples in machine learning, Artificial Intelligence 97 (1997), 245–271.

    MATH  Article  MathSciNet  Google 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.

  8. [8]

    P. Cameron: Combinatorics: topics, techniques, algorithms; Cambridge University Press, 1994.

  9. [9]

    D. Helmbold, R. Sloan and M. Warmuth: Learning integer lattices, SIAM Journal of Computing 21(2) (1992), 240–266.

    MATH  Article  MathSciNet  Google 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.

    MATH  Article  MathSciNet  Google 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.

  12. [12]

    A. Kumchev: The distribution of prime numbers, manuscript, 2005.

  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.

    MATH  Article  MathSciNet  Google 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.

  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.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    E. Mossel, R. O’Donnell and R. Servedio: Learning juntas, in: STOC, pages 206-212, 2003.

  17. [17]

    G. Pólya and G. Szegő: Problems and theorems in Analysis, II; Springer, 1976.

  18. [18]

    T. Siegenthaler: Correlation-immunity of nonlinear combining functions for cryptographic applications, IEEE Transactions on Information Theory 30(5) (1984), 776–780.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    L. Valiant: A theory of the learnable, Communications of the ACM 27(11) (1984), 1134–1142.

    MATH  Article  Google 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.

  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.

  22. [22]

    J. von zur Gathen and J. Roche: Polynomials with two values, Combinatorica 17(3) (1997), 345–362.

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Evangelos Markakis.

Additional information

This work was done when all authors were at the Georgia Institute of Technology and it is based on the preliminary versions [14] and [11].

Partially supported by European Commission IHP Network HARP (Harmonic Analysis and Related Problems), Contract Number: HPRN-CT-2001-00273 — HARP, and by grant INTAS 03-51-5070 (2004) (Analytical and Combinatorial Methods in Number Theory and Geometry).

Research supported by NSF grant CCF-0431023.

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Kolountzakis, M.N., Lipton, R.J., Markakis, E. et al. On the Fourier spectrum of symmetric Boolean functions. Combinatorica 29, 363–387 (2009). https://doi.org/10.1007/s00493-009-2310-z

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Mathematics Subject Classification (2000)

  • 42B05
  • 68Q32