We prove two main results on how arbitrary linear threshold functions over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold ...
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We prove two main results on how arbitrary linear threshold functions over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is -close to a threshold function depending only on many variables, where denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut's well-known theorem (Friedgut in Combinatorica 18(1):474-483, 1998), which states that every Boolean function f is -close to a function depending only on many variables, for the case of threshold functions. We complement this upper bound by showing that many variables are required for -approximating threshold functions. Our second result is a proof that every n-variable threshold function is -close to a threshold function with integerweights at most This is an improvement, in the dependence on the error parameter , on an earlier result of Servedio (Comput Complex 16(2):180-209, 2007) which gave a bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original result of Servedio (Comput Complex 16(2):180-209, 2007) and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.
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