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)

On the size of homogeneous and of depth four formulas with low individual degree

International audienceLet r ≥ 1 be an integer. Let us call a polynomial f (x_1,...,x_N) ∈ F[x] as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. We prove lower bounds for several subclasses of such circuits. Specifically, first define the formal degree of a node α with respect to a variable x_i inductively as follows. For a leaf α it is 1 if α is labelled with x_i and zero otherwise; for an internal node α labelled with × (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x_i. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits

Shallow Circuits with High-Powered Inputs

A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for (high-degree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (low-degree) multivariate identity testing are weaker. To obtain our lower bound it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the Shub-Smale tau-conjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a good enough bound on the number of real roots of sums of products of sparse polynomials (Descartes' rule of signs gives such a bound for sparse polynomials and products thereof). In this third version of our paper we show that the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte

Towards Optimal Depth Reductions for Syntactically Multilinear Circuits

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 ($\Sigma\Pi\Sigma\Pi$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = \omega(n/\log n)$, 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 $d < n^{\epsilon}$ for a small enough constant $\epsilon$. Our upper bound matches the lower bound of $\exp\left({\Omega\left(\sqrt{n\log n}\right)}\right)$ 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 $\Delta$ of size at most $\exp\left(O\left(\Delta \cdot (n/\log n)^{1/\Delta} \cdot \log n\right)\right)$. It follows from the lower bounds of Raz and Yehudayoff (CC 2009) that in general, for constant $\Delta$, the exponent in this upper bound is tight and cannot be improved to $o\left(\left(n/\log n\right)^{1/\Delta}\cdot \log n\right)$

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 $GapL$

Sum of Products of Read-Once Formulas

We study limitations of polynomials computed by depth two circuits built over read-once formulas (ROFs). In particular, 1. We prove an exponential lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [CC,2009]. 2. We obtain an exponential lower bound on the size of arithmetic circuits computing sum of products of restricted ROFs of unbounded depth computing the permanent of an n by n matrix. The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by sqrt(n). Additionally, we restrict the product fan in to be bounded by a sub linear function. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial. 3. We also show an exponential lower bound for the above model against a polynomial in VP. 4. Finally we observe that the techniques developed yield an exponential lower bound on the size of sums of products of syntactically multilinear arithmetic circuits computing a product of variable disjoint linear forms where the bottom sum gate and product gates at the second level have fan in bounded by a sub linear function. Our proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [STOC 2004]

An Almost Cubic Lower Bound for Depth Three Arithmetic Circuits

We show an almost cubic lower bound on the size of any depth three arithmetic circuit computing an explicit multilinear polynomial in n variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson [CCC, 1999]