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\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/2}))}\) for depth-3 multilinear formulas, for some explicit polynomial.For depth-4 multilinear formulas, of size exp\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/4}))}\) 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\({\left(n^{1- \delta}\right)}\), for regular depth-d multilinear formulas with formal degree at most n and size exp\({(n^\delta)}\), where \({\delta = O(1/{\sqrt{5}^d})}\). This result implies a lower bound of roughly exp\({(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}\) 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).