Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

In this paper, we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds, we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp(n^δ) , we give a hitting set of size exp(O~ (n^{2 / 3 + 2 δ / 3})). This implies a lower bound of exp(Ω ~ (n^{1 / 2})) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(n^δ) , we give a hitting set of size exp(O~ (n^{2 / 3 + 4 δ / 3})). This implies a lower bound of exp(Ω ~ (n^{1 / 4})) for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of + , × gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp(n^{1 - δ}) , for regular depth-d multilinear formulas with formal degree at most n and size exp(n^δ) , where δ=O(1/5d). This result implies a lower bound of roughly exp(Ω~(n1/5d)) for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas, we go straight to read-once algebraic branching programs).