Average-Case Rigidity Lower Bounds

Abstract

It is shown that there exists f: { 0, 1 } n/2× { 0, 1 } n/2→ { 0, 1 } in E NP such that for every 2 n/2× 2 n/2 matrix M of rank ≤ ρ we have Px,y[ f(x, y) ≠ Mx,y] ≥ 1 / 2 - 2 -Ω(k), whenever log ρ≤ δn/ k(log n+ k) for a sufficiently small δ> 0, and n is large enough. This generalizes recent results which bound below the probability by 1 / 2 - Ω(1 ) or apply to constant-depth circuits.