, 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


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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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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