Quasi-polynomial Hitting-set for Set-depth-Delta Formulas

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition X1 t · · · t Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1 ∏d j=1 fi,j(xXj ), where fi,j is a sparse polynomial in F[xXj ]. Extending this definition to any depth we call a depth-∆ formula C (consisting of alternating layers of Σ and Π gates, with a Σ-gate on top) a set-depth-∆ formula if every Π-layer in C respects a (unknown) partition on the variables; if ∆ is even then the product gates of the bottom-most Π-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-∆ formulas (over any field) with running time polynomial in exp((∆ log s)∆−1), where s is the size bound on the input set-depth-∆ formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of ∆ = 3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth log s/ log log s, for a fixed constant < 1. The technique is to investigate depth-∆ formulas via depth-(∆ − 1) formulas over a Hadamard algebra, after applying a ‘shift’ on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-∆ formulas.