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Spectral Norm of Symmetric Functions

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2012, RANDOM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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 {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.

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Authors and Affiliations

  1. School of Computer Science, McGill University, Canada

    Anil Ada, Omar Fawzi & Hamed Hatami

Authors
  1. Anil Ada
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  2. Omar Fawzi
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  3. Hamed Hatami
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, Gates Building 7203, 15213, Pittsburgh, PA, USA

    Anupam Gupta

  2. Department of Computer Science, University Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Centre Universitaire d’Informatique, University of Geneva, Battelle A, 7 route de Drize, 1227, Carouge, Switzerland

    José Rolim

  4. Department of Computer Science, Foundation School of Engineering and Applied Science, Columbia University, Amsterdam Avenue 1214, 10027-7003, New York, NY, USA

    Rocco Servedio

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© 2012 Springer-Verlag Berlin Heidelberg

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Ada, A., Fawzi, O., Hatami, H. (2012). Spectral Norm of Symmetric Functions. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_29

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  • DOI: https://doi.org/10.1007/978-3-642-32512-0_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32511-3

  • Online ISBN: 978-3-642-32512-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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