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Abstract

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.

Keywords

Boolean Function Symmetric Function Fourier Spectrum Communication Complexity Full Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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