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Abstract

We consider the problem of testing whether a Boolean function f:{ − 1,1} n  → { − 1,1} is a ±1-weight halfspace, i.e. a function of the form f(x) = sgn(w 1 x 1 + w 2 x 2 + ⋯ + w n x n ) where the weights w i take values in { − 1,1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±-weight halfspace versus ε-far from all such halfspaces we prove that nonadaptive algorithms must make Ω(logn) queries. We complement this lower bound with a sublinear upper bound showing that \(O(\sqrt{n}\cdot \)poly\((\frac{1}{\epsilon}))\) queries suffice.

Keywords

Boolean Function Testing Algorithm Random Index Standard Gaussian Random Variable Boolean Cube 
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 2009

Authors and Affiliations

  • Kevin Matulef
    • 1
  • Ryan O’Donnell
    • 2
  • Ronitt Rubinfeld
    • 3
  • Rocco A. Servedio
    • 4
  1. 1.MITUSA
  2. 2.Carnegie Mellon UniversityUSA
  3. 3.Tel Aviv University and MITIsrael
  4. 4.Columbia UniversityUSA

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