The spectral norm of a Boolean function f:{0,1} n  → { − 1,1} is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as r(f)log(n/r(f)) where r(f) =  max {r 0,r 1}, and r 0 and r 1 are the smallest integers less than n/2 such that f(x) or \(f(x) \cdot \textnormal{\textsc{parity}}(x)\) is constant for all x with ∑ x i  ∈ [r 0, n − r 1]. We mention some applications to the decision tree and communication complexity of symmetric functions.


Boolean Function Symmetric Function Fourier Spectrum Communication Complexity Full Version 
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  1. 1.
    Ada, A., Fawzi, O., Hatami, H.: Spectral norm of symmetric functions. arXiv:1205.5282 (2012)Google Scholar
  2. 2.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bruck, J., Smolensky, R.: Polynomial threshold functions, ac0 functions, and spectral norms. SIAM J. Comput. 21(1), 33–42 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    de Wolf, R.: A note on quantum algorithms and the minimal degree of ε-error polynomials for symmetric functions. Quantum Inf. Comput. 8(10), 943–950 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Goldmann, M., Håstad, J., Razborov, A.A.: Majority gates vs. general weighted threshold gates. Comput. Complex., 277–300 (1992)Google Scholar
  6. 6.
    Green, B., Sanders, T.: Boolean functions with small spectral norm. Geom. Funct. Anal. 18(1), 144–162 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Grolmusz, V.: On the power of circuits with gates of low l1 norms. Theor. Comput. Sci. 188(1-2), 117–128 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Grolmusz, V.: Harmonic analysis, real approximation, and the communication complexity of boolean functions. Algorithmica 23(4), 341–353 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kalai, A.T., Klivans, A.R., Mansour, Y., Servedio, R.A.: Agnostically learning halfspaces. SIAM J. Comput. 37(6), 1777–1805 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Klivans, A.R., Sherstov, A.A.: Lower bounds for agnostic learning via approximate rank. Comput. Complex. 19(4), 581–604 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kolountzakis, M.N., Lipton, R.J., Markakis, E., Mehta, A., Vishnoi, N.K.: On the fourier spectrum of symmetric boolean functions. Combinatorica 29(3), 363–387 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kushilevitz, E., Mansour, Y.: Learning decision trees using the fourier spectrum. In: Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, STOC 1991, pp. 455–464. ACM, New York (1991)CrossRefGoogle Scholar
  13. 13.
    Lee, T., Shraibman, A.: Lower bounds in communication complexity, vol. 3. Now Publishers Inc. (2009)Google Scholar
  14. 14.
    O’Donnell, R., Servedio, R.A.: Extremal properties of polynomial threshold functions. J. Comput. Syst. Sci. 74(3), 298–312 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    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, Part I. LNCS, vol. 6755, pp. 330–341. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Paturi, R.: On the degree of polynomials that approximate symmetric Boolean functions (preliminary version). In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp. 468–474. ACM, New York (1992)CrossRefGoogle Scholar
  17. 17.
    Razborov, A.: Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics 67(1), 145–159 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Sherstov, A.A.: Approximate inclusion-exclusion for arbitrary symmetric functions. Comput. Complex. 18(2), 219–246 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sherstov, A.A.: The pattern matrix method. SIAM J. Comput. 40(6), 1969–2000 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Shi, Y., Zhang, Z.: Communication complexities of symmetric XOR functions. Quantum Inf. Comput. (available at arXiv:0808.1762) 9, 255–263 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Shpilka, A., Tal, A.: On the minimal fourier degree of symmetric boolean functions. In: Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity, CCC 2011, pp. 200–209. IEEE Computer Society, Washington, DC (2011)CrossRefGoogle Scholar
  22. 22.
    Siu, K.-Y., Bruck, J.: On the power of threshold circuits with small weights. SIAM J. Discrete Math. 4(3), 423–435 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anil Ada
    • 1
  • Omar Fawzi
    • 1
  • Hamed Hatami
    • 1
  1. 1.School of Computer ScienceMcGill UniversityCanada

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