Property Testing pp 334-340

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6390)

Testing (Subclasses of) Halfspaces

  • Kevin Matulef
  • Ryan O’Donnell
  • Ronitt Rubinfeld
  • Rocco Servedio

Abstract

We address the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w . x − θ). We consider halfspaces over the continuous domain Rn (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube { − 1,1}n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are ε-far from any halfspace using only poly\((\frac{1}{\epsilon})\) queries, independent of the dimension n.

In contrast to the case of general halfspaces, we show that testing natural subclasses of halfspaces can be markedly harder; for the class of { − 1,1}-weight halfspaces, we show that a tester must make at least Ω(logn) queries. We complement this lower bound with an upper bound showing that \(O(\sqrt{n})\) queries suffice.

Keywords

halfspaces linear thresholds functions 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Kevin Matulef
    • 1
  • Ryan O’Donnell
    • 2
  • Ronitt Rubinfeld
    • 3
  • Rocco Servedio
    • 4
  1. 1.ITCS, Tsinghua UniversityChina
  2. 2.Carnegie Mellon UniversityUSA
  3. 3.Massachusetts Institute of TechnologyUSA
  4. 4.Columbia UniversityUSA

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