  We consider statistical estimations of a matrix product over the integers in
a distributed setting, where we have two parties Alice and Bob; Alice holds a
matrix $A$ and Bob holds a matrix $B$, and they want to estimate statistics of
$A \cdot B$. We focus on the well-studied $\ell_p$-norm, distinct elements ($p
= 0$), $\ell_0$-sampling, and heavy hitter problems. The goal is to minimize
both the communication cost and the number of rounds of communication.
  This problem is closely related to the fundamental set-intersection join
problem in databases: when $p = 0$ the problem corresponds to the size of the
set-intersection join. When $p = \infty$ the output is simply the pair of sets
with the maximum intersection size. When $p = 1$ the problem corresponds to the
size of the corresponding natural join. We also consider the heavy hitters
problem which corresponds to finding the pairs of sets with intersection size
above a certain threshold, and the problem of sampling an intersecting pair of
sets uniformly at random.
