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.


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