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A bound on the precision required to estimate a boolean perceptron from its average satisfying assignment
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2006
"... A boolean perceptron is a linear threshold function over the discrete boolean domain f0; 1g n That is, it maps any binary vector to 0 or 1 depending on whether the vector's components satisfy some linear inequality. In 1961, Chow showed that any boolean perceptron is determined by the average or &q ..."
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A boolean perceptron is a linear threshold function over the discrete boolean domain f0; 1g n That is, it maps any binary vector to 0 or 1 depending on whether the vector's components satisfy some linear inequality. In 1961, Chow showed that any boolean perceptron is determined by the average or "center of gravity " of its "true " vectors (those that are mapped to 1), together with the total number of true vectors. Moreover, these quantities distinguish the function from any other boolean function, not just other boolean perceptrons. In this paper we go further, by identifying a lower bound on the Euclidean distance between the average satisfying assignment of a boolean perceptron, and the average satisfying assignment of a boolean function that disagrees with that boolean perceptron on a fraction ffl of the input vectors. The distance between the two means is shown to be at least (ffl=n) O(log(n=ffl) log(1=ffl)) This is motivated by the statistical question of whether an empirical estimate of this average allows us to recover a good approximation to the perceptron. Our result provides a mildly superpolynomial upper bound on the growth rate of the sample size required to learn boolean perceptrons in the "restricted focus of attention " setting. In the process we also find some interesting geometrical properties of the vertices of the unit hypercube.
Estimating a Boolean Perceptron from its Average Satisfying Assignment: A Bound on the Precision Required
 In Proceedings of the Fourteenth Annual Conference on Computational Learning Theory
, 2001
"... . A boolean perceptron is a linear threshold function over the discrete boolean domain f0; 1g n . That is, it maps any binary vector to 0 or 1 depending on whether the vector's components satisfy some linear inequality. In 1961, Chow [9] showed that any boolean perceptron is determined by the ..."
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. A boolean perceptron is a linear threshold function over the discrete boolean domain f0; 1g n . That is, it maps any binary vector to 0 or 1 depending on whether the vector's components satisfy some linear inequality. In 1961, Chow [9] showed that any boolean perceptron is determined by the average or \center of gravity" of its \true" vectors (those that are mapped to 1). Moreover, this average distinguishes the function from any other boolean function, not just other boolean perceptrons. We address an associated statistical question of whether an empirical estimate of this average is likely to provide a good approximation to the perceptron. In this paper we show that an estimate that is accurate to within additive error (=n) O(log(1=)) determines a boolean perceptron that is accurate to within error (the fraction of misclassied vectors). This provides a mildly superpolynomial bound on the sample complexity of learning boolean perceptrons in the \restricted focus of attention" setting. In the process we also nd some interesting geometrical properties of the vertices of the unit hypercube. 1