Testing (Subclasses of) Halfspaces
scopus(2010)
摘要
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
(\frac1e)(\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(Ön)O(\sqrt{n}) queries suffice.
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关键词
linear thresholds functions.,dimension n,general halfspaces,uniform distribution,standard multivariate gaussian distribution,boolean-valued function,n queries suffice,halfspaces,distinguishes halfspaces,upper bound showing,log n,weight halfspaces,upper bound,lower bound,gaussian distribution,value function
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