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Discrete Mathematics
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Computational Complexity
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SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
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FOCS '61 Proceedings of the 2nd Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 1961)
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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(wċx-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(1/ε) 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 O(log n) queries. We complement this lower bound with an upper bound showing that O(√n queries suffice.