Property Testing

Volume 6390 of the series Lecture Notes in Computer Science pp 334-340

Testing (Subclasses of) Halfspaces

  • Kevin MatulefAffiliated withITCS, Tsinghua University
  • , Ryan O’DonnellAffiliated withCarnegie Mellon University
  • , Ronitt RubinfeldAffiliated withMassachusetts Institute of Technology
  • , Rocco ServedioAffiliated withColumbia University

* Final gross prices may vary according to local VAT.

Get Access


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 R n (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.


halfspaces linear thresholds functions