The Sign-Rank of AC^O

The sign-rank of a matrix A = [A_ij] with plusmn1 entries is the least rank of a real matrix B = [B_ij] with A_ijB_ij > 0 for all i, j. We obtain the first exponential lower bound on the sign-rank of a function in AC⁰. Namely, let f(x, y) = Lambda_i=1 ^m Lambda_j=1 ^{m 2}(x_ij Lambda y_ij). We show that the matrix [f(x, y)]_{x, y} has sign-rank 2^Omega(m). This in particular implies that Sigma₂ ^ccnsubeUPP^cc, which solves a long-standing open problem posed by Babai, Frankl, and Simon (1986). Our result additionally implies a lower bound in learning theory. Specifically, let Phi₁,..., Phi_r : {0, 1}ⁿ rarrRopf be functions such that every DNF formula f : {0, 1}ⁿ rarr {-1, +1} of polynomial size has the representation f equiv sign(a₁Phi₁ + hellip + a_rPhi_r) for some reals a₁,..., a_r. We prove that then r ges 2^{Omega(n 1/3 )}, which essentially matches an upper bound of 2^{Otilde(n 1/3 )} due to Klivans and Servedio (2001). Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in AC⁰. This substantially generalizes and strengthens the results of Krause and Pudlak (1997).