Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0
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The threshold degree of a Boolean function f{0,1}^n→{0,1} is the minimum degree of a real polynomial p that represents f in sign: sgn p(x)=(-1)^f(x). A related notion is sign-rank, defined for a Boolean matrix F=[F_ij] as the minimum rank of a real matrix M with sgn M_ij=(-1)^F_ij. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (AC^0) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications. We give an essentially optimal solution to this problem. For any ϵ>0, we construct an AC^0 circuit in n variables that has threshold degree Ω(n^1-ϵ) and sign-rank (Ω(n^1-ϵ)), improving on the previous best lower bounds of Ω(√(n)) and (Ω̃(√(n))), respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of AC^0 circuits of any given depth, with a strict improvement starting at depth 4. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of AC^0, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of AC^0.
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