On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits

International audienceIn this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree of r (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better and better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing a multilinear polynomial on n^O(1) variables and degree d = o(n), must have size at least (n/r^1.1)^{\sqrt{d/r}} when r is o(d) and is strictly less than n^1/1.1. This bound however deteriorates with increasing r. It is a natural question to ask if we can prove a bound that does not deteriorate with increasing r or a bound that holds for a larger regime of r. We here prove a lower bound which does not deteriorate with r , however for a specific instance of d = d (n) but for a wider range of r. Formally, we show that there exists an explicit polynomial on n^{O(1)} variables and degree Θ(log^2(n)) such that any depth four circuit of bounded individual degree r < n^0.2 must have size at least exp(Ω (log^2 n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017)

Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach

Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to prove VP not equal to VNP, it is sufficient to show that an explicit polynomial in VNP of degree n requires 2^{\omega(\sqrt{n}\log n)} size depth-4 circuits. Soon after Tavenas's result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2^{\Omega(\sqrt{n}\log n)} have been proved Kayal et al. and Fournier et al. In particular, using combinatorial design Kayal et al.\ construct an explicit polynomial in VNP that requires depth-4 circuits of size 2^{\Omega(\sqrt{n}\log n)} and Fournier et al.\ show that iterated matrix multiplication polynomial (which is in VP) also requires 2^{\Omega(\sqrt{n}\log n)} size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies the property would achieve similar circuit size lower bound for depth-4 circuits. In particular, it does not matter whether f is in VP or in VNP. As a result, we get a very simple unified lower bound analysis for the above mentioned polynomials. Another goal of this paper is to compare between our current knowledge of depth-4 circuit size lower bounds and determinantal complexity lower bounds. We prove the that the determinantal complexity of iterated matrix multiplication polynomial is \Omega(dn) where d is the number of matrices and n is the dimension of the matrices. So for d=n, we get that the iterated matrix multiplication polynomial achieves the current best known lower bounds in both fronts: depth-4 circuit size and determinantal complexity. To the best of our knowledge, a \Theta(n) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for constant d>1 by Jansen.Comment: Extension of the previous uploa

Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then $C$ must have size $2^{\Omega(\sqrt{d}\log{d})}$. The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for $ACC^0$ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in $ACC^0$ can also be computed by algebraic $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits (i.e., circuits of the form -- sums of powers of polynomials) of $2^{\log^{O(1)}n}$ size. Thus they argued that a $2^{\omega(\log^{O(1)}{n})}$ "functional" lower bound for an explicit polynomial $Q$ against $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits would imply a lower bound for the "corresponding Boolean function" of $Q$ against non-uniform $ACC^0$. In their work, they ask if their lower bound be extended to $\Sigma\mathord{\wedge}\Sigma\Pi$ circuits. In this paper, for large integers $n$ and $d$ such that $\omega(\log^2n)\leq d\leq n^{0.01}$, we show that any $\Sigma\mathord{\wedge}\Sigma\Pi$ circuit of bounded individual degree at most $O\left(\frac{d}{k^2}\right)$ that functionally computes Iterated Matrix Multiplication polynomial $IMM_{n,d}$ ($\in VP$) over $\{0,1\}^{n^2d}$ must have size $n^{\Omega(k)}$. Since Iterated Matrix Multiplication $IMM_{n,d}$ over $\{0,1\}^{n^2d}$ is functionally in $GapL$, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of $ACC^0$ from $GapL$

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting. We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(\Delta)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $\Delta+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $\Delta$ and size less than $\exp(n^{\Omega(1/\Delta)})$. This result is tight up to the constant implicit in the double exponent for all $\Delta = o(\log n/\log \log n).$ 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 $\Delta = 1.$ 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

Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications

The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within P. In this paper, we study the algebraic formula complexity of multiplying d many 2x2 matrices, denoted IMM_d, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear. Formally, for each depth Delta <= log d, we show that any product-depth Delta multilinear formula for IMM_d must have size exp(Omega(Delta d^{1/Delta})). It also follows from this that any multilinear circuit of product-depth Delta for the same polynomial of the above form must have a size of exp(Omega(d^{1/Delta})). In particular, any polynomial-sized multilinear formula for IMM_d must have depth Omega(log d), and any polynomial-sized multilinear circuit for IMM_d must have depth Omega(log d/log log d). Both these bounds are tight up to constant factors. Our lower bound has the following consequences for multilinear formula complexity. Depth-reduction: A well-known result of Brent (JACM 1974) implies that any formula of size s can be converted to one of size s^{O(1)} and depth O(log s); further, this reduction continues to hold for multilinear formulas. On the other hand, our lower bound implies that any depth-reduction in the multilinear setting cannot reduce the depth to o(log s) without a superpolynomial blow-up in size. Separations from general formulas: Shpilka and Yehudayoff (FnTTCS 2010) asked whether general formulas can be more efficient than multilinear formulas for computing multilinear polynomials. Our result, along with a non-trivial upper bound for IMM_d implied by a result of Gupta, Kamath, Kayal and Saptharishi (SICOMP 2016), shows that for any size s and product-depth Delta = o(log s), general formulas of size s and product-depth Delta cannot be converted to multilinear formulas of size s^{O(1)} and product-depth Delta, when the underlying field has characteristic zero

A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas

We show explicit separations between the expressive powers of multilinear formulas of small-depth and all polynomial sizes. Formally, for any s = s(n) = n^{O(1)} and any delta>0, we construct explicit families of multilinear polynomials P_n in F[x_1,...,x_n] that have multilinear formulas of size s and depth three but no multilinear formulas of size s^{1/2-delta} and depth o(log n/log log n). As far as we know, this is the first such result for an algebraic model of computation. Our proof can be viewed as a derandomization of a lower bound technique of Raz (JACM 2009) using epsilon-biased spaces