Abstract
A real multivariate polynomial p(x 1, …, x n ) is said to sign-represent a Boolean function f: {0,1}n →{−1,1} if the sign of p(x) equals f(x) for all inputs x∈{0,1}n . We give new upper and lower bounds on the degree of polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of superconstant depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved constructively; we give explicit dual solutions to the necessary linear programs.
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