Lecture Notes in Computer Science Volume 6390, 2010, pp 334-340

Testing (Subclasses of) Halfspaces

Purchase on Springer.com

$29.95 / €24.95 / £19.95*

* Final gross prices may vary according to local VAT.

Get Access

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 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.