We present a randomized algorithm for reconstructing multilinear depth-4 arithmetic circuits with fan-in 2 at the top + gate. The algorithm is given blackbox access to a multilinear polynomial f in F[x_1,..,x_n] computable by a multilinear Sum-Product-Sum-Product(SPSP) circuit of size s and outputs an equivalent multilinear SPSP circuit, runs ... more >>>

We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas. That is, we give
an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with $0, 1$-coefficients such that
for any representation of ... more >>>

We consider the \emph{black-box} polynomial identity testing problem for a sub-class of
depth-4 circuits. Such circuits compute polynomials of the following type:
$
C(x) = \sum_{i=1}^k \prod_{j=1}^{d_i} Q_{i,j},
$
where $k$ is the fan-in of the top $\Sigma$ gate and $r$ is the maximum degree of the ... more >>>

In 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 ... more >>>

Kayal, Saha and Tavenas [Theory of Computing, 2018] showed that for all large enough integers $n$ and $d$ such that $d\geq \omega(\log{n})$, any syntactic depth four circuit of bounded individual degree $\delta = o(d)$ that computes the Iterated Matrix Multiplication polynomial ($IMM_{n,d}$) must have size $n^{\Omega\left(\sqrt{d/\delta}\right)}$. Unfortunately, this bound ... more >>>

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 ... more >>>