Combinatorica

, Volume 29, Issue 3, pp 363–387

On the Fourier spectrum of symmetric Boolean functions

Authors

  • Mihail N. Kolountzakis
    • Department of MathematicsUniv. of Crete
  • Richard J. Lipton
    • Georgia TechCollege of Computing
    • Telcordia Research
    • Centre for Math and Computer Science (CWI)
  • Aranyak Mehta
    • IBM Almaden Research Center
  • Nisheeth K. Vishnoi
    • College of ComputingGeorgia Institute of Technology
    • IBM India Research Lab
Article

DOI: 10.1007/s00493-009-2310-z

Cite this article as:
Kolountzakis, M.N., Lipton, R.J., Markakis, E. et al. Combinatorica (2009) 29: 363. doi:10.1007/s00493-009-2310-z

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 no(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 n2κ/3.

Mathematics Subject Classification (2000)

42B0568Q32

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2009