We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition
(X_1,...,X_d) of the variable indices [n] that the top product layer respects,
i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a
sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we
call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi
gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in
C respects a (unknown) partition on the variables; if Delta is even then the
product gates of the bottom-most Pi-layer are allowed to compute arbitrary
monomials.
In this work, we give a hitting-set generator for set-depth-Delta formulas
(over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta -
1}), where s is the size bound on the input set-depth-Delta 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 Delta=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 epsilon.log s / loglog s, for a fixed constant epsilon < 1.
The technique is to investigate depth-Delta formulas via depth-(Delta-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-Delta
formulas.Comment: 22 page
