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We consider statistical estimations of a matrix product over the integers ina distributed setting, where we have two parties Alice and Bob; Alice holds amatrix $A$ 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 minimizeboth the communication cost and the number of rounds of communication.  This problem is closely related to the fundamental set-intersection joinproblem in databases: when $p = 0$ the problem corresponds to the size of theset-intersection join. When $p = \infty$ the output is simply the pair of setswith the maximum intersection size. When $p = 1$ the problem corresponds to thesize of the corresponding natural join. We also consider the heavy hittersproblem which corresponds to finding the pairs of sets with intersection sizeabove a certain threshold, and the problem of sampling an intersecting pair ofsets uniformly at random.

Distributed Statistical Estimation of Matrix Products with Applications