We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$, $q=1$), and has been ... more >>>

Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC'18) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few ... more >>>

We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One only need to consider size-$s$ degree-$s$ circuits that depend on the first $\log^{\circ c} s$ variables (where $c$ is a constant and we are ... more >>>