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 >>>

A circuit $C$ \emph{compresses} a function $f:\{0,1\}^n\rightarrow \{0,1\}^m$ if given an input $x\in \{0,1\}^n$ the circuit $C$ can shrink $x$ to a shorter $\ell$-bit string $x'$ such that later, a computationally-unbounded solver $D$ will be able to compute $f(x)$ based on $x'$. In this paper we study the existence of ... more >>>

We show that deterministic communication complexity can be superlogarithmic in the partition number of the associated communication matrix. We also obtain near-optimal deterministic lower bounds for the Clique vs. Independent Set problem, which in particular yields new lower bounds for the log-rank conjecture. All these results follow from a simple ... more >>>