We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($ΣΠΣΠ$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = ω(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas~\cite{T15} for general circuits, and is known to be asymptotically optimal in the exponent when $d < n^ε$ for a small enough constant $ε$. Our upper bound matches the lower bound of $\exp\left({Ω\left(\sqrt{n\log n}\right)}\right)$ proved by Raz and Yehudayoff~\cite{RY09}, and thus cannot be improved further in the exponent. Our results hold over all fields and also generalize to circuits of small individual degree.
More generally, we show that an $n$-variate polynomial computable by a syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a syntactically multilinear circuit of product-depth $Δ$ of size at most $\exp\left(O\left(Δ\cdot (n/\log n)^{1/Δ} \cdot \log n\right)\right)$. It follows from the lower bounds of Raz and Yehudayoff (CC 2009) that in general, for constant $Δ$, the exponent in this upper bound is tight and cannot be improved to $o\left(\left(n/\log n\right)^{1/Δ}\cdot \log n\right)$.
In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes,Wigderson and Yehudayoff [HWY10]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, the…
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…
We prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial $f$ in VNP defined over $n^2$ variables, and of degree $n$, such that any product-depth $Δ$ set-multilinear formula computing $f$ has size at least $n^{Ω\left( n^{1/Δ}/Δ\right)}$. The hard polynomial $f$ comes from the class of Nisan-Wigderson (NW) design-based polynomials. Our lower bounds improve upon the recent work of Limaye, Srinivasan and Tavenas (STOC 2022), where a lower bound of the form $(\log n)^{Ω(Δn^{1/Δ})}$ was shown for the …
Algebraic Branching Programs(ABPs) are standard models for computing polynomials. Syntactic multilinear ABPs (smABPs) are restrictions of ABPs where every variable is allowed to occur at most once in every path from the start to the terminal node. Proving lower bounds against syntactic multilinear ABPs remains a challenging open question in Algebraic Complexity Theory. The current best known bound is only quadratic [Alon-Kumar-Volk, ECCC 2017]. In this article we develop a new approach upper bounding the rank of the partial derivative matrix of syntactic multlinear ABPs: Convert the ABP to a s…
We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant $\sum_{i,j,k,\ell\in[n]} z_{ijk\ell}x_ix_j x_k x_\ell - β=0$, for Boolean variables, does not have polynomial-size IPS refutations where the refutations are multilinear and written as constant-depth circuits. Andrews and Forbes (STOC'22) established recently a constant-depth IPS lower bound, but their hard instance does not have itself small constant-depth circuits, while our inst…
We give new and efficient black-box reconstruction algorithms for some classes of depth-$3$ arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a constant-rank tensor. More specifically, we provide efficient learning algorithms that run in randomized polynomial time over general fields and in deterministic polynomial time over the reals and the complex numbers for the following classes: (1) Set-multilinear depth-$3$ circuits of constant top f…
We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving lower bounds for low-depth set-multilinear circuits. An interesting aspect of our proof is that it does not require conversion of a circuit to a set-multilinear circuit, nor does it involve a random restrictio…
