We study the size blow-up that is necessary to convert an algebraic circuit of product-depth δ+1 to one of product-depth δ in the multilinear setting. We show that for every positive δ = δ(n) = o(log n log log n), there is an explicit multilinear polynomial P^(δ) on n variables that can be computed by a multilinear formula of product-depth δ+1 and size O(n), but not by any multilinear circuit of product-depth δ and size less than exp(n^Ω(1 δ)). This result is tight up to the constant implicit in the double exponent for all δ = o(log n log log n). This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case δ = 1. Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits.
When the algebraic circuits are instead restricted to be multilinear, more is known. Raz and Yehudayoff @cite_7 were the first to study this question, and showed a quasipolynomial depth-hierarchy theorem for all constant product-depths. In the case of product-depth @math vs. product-depth @math , this was strengthened to an exponential separation by Kayal, Nair and Saha @cite_37 .
@cite_37 @cite_7
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We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following: 1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^ Omega(n) size. 2. Any multilinear depth three circuit computing IMM_ n,d (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^ Omega(d) size. IMM_ n,d can be easily computed by a poly(n,d) sized ROABP. 3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^ Omega(d) . This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field. We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth d and product-depth d+1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that (1) There exists a multilinear circuit of product-depth d+1 and of polynomial size computing f and (2) Every multilinear circuit of product-depth d computing f has super-polynomial size.