Improved Bounds on the Sign-Rank of AC

The sign-rank of a matrix A with entries in {−1,+1} is the least rank of a real matrix B with Aij · Bij > 0 for all i, j. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC, answering an old question of Babai, Frankl, and Simon (1986). Specifically, they exhibited a matrix A = [F (x, y)]x,y for a specific function F : {−1, 1} × {−1, 1} → {−1, 1} in AC, such that A has sign-rank exp(Ω(n)). We prove a generalization of Razborov and Sherstov’s result, yielding exponential sign-rank lower bounds for a non-trivial class of functions (that includes the function used by Razborov and Sherstov). As a corollary of our general result, we improve Razborov and Sherstov’s lower bound on the sign-rank of AC from exp(Ω(n)) to exp(Ω̃(n)). We also describe several applications to communication complexity, learning theory, and circuit complexity.