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.
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Matulef, K., O’Donnell, R., Rubinfeld, R., Servedio, R. (2010). Testing (Subclasses of) Halfspaces. In: Goldreich, O. (eds) Property Testing. Lecture Notes in Computer Science, vol 6390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16367-8_27
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DOI: https://doi.org/10.1007/978-3-642-16367-8_27
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