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$${(n^\delta)}$$(nδ), we give a hitting set of size exp$${\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}$$O~n2/3+2δ/3. This implies a lower bound of exp$${(\tilde{\Omega}(n^{1/2}))}$$(Ω~(n1/2)) for depth-3 multilinear formulas, for some explicit polynomial.For depth-4 multilinear formulas, of size exp$${(n^\delta)}$$(nδ), we give a hitting set of size exp$${\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}$$O~n2/3+4δ/3. This implies a lower bound of exp$${(\tilde{\Omega}(n^{1/4}))}$$(Ω~(n1/4)) for depth-4 multilinear formulas, for some explicit polynomial.A regular formula consists of alternating layers of $${+,\times}$$+,× gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp$${\left(n^{1- \delta}\right)}$$n1-δ, for regular depth-d multilinear formulas with formal degree at most n and size exp$${(n^\delta)}$$(nδ), where $${\delta = O(1/{\sqrt{5}^d})}$$δ=O(1/5d). This result implies a lower bound of roughly exp$${(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}$$(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).