Spectral Norm of Symmetric Functions

  • Anil Ada
  • Omar Fawzi
  • Hamed Hatami
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

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 {r0,r1}, and r0 and r1 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 ∑ xi ∈ [r0, n − r1]. We mention some applications to the decision tree and communication complexity of symmetric functions.

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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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