Let $A \in R^{n×n}$ and $B \in R^{n×n}$ be two matrices. Let $s$ be some positive integer. Let $\tilde{A} \in R^{n×n}$ be a random matrix with mutually independent entries:
$$ \tilde{a}_{ij} = \begin{cases} \frac{a_{ij}}{p_{ij}}, & \text{with prob. $p_{ij}$} \\ 0, & \text{with prob. $1 - p_{ij}$} \end{cases}$$
where $p_{ij}$ are some parameters depending on $A$ and $s$ such that $\sum_{ij}p_{ij} \le sn$. Show that for some choice of parameters $p_{ij}$, $$E[\mid\mid \tilde{A}B - AB \mid\mid^{2}_{F}] \le \frac{1}{s}\mid\mid A\mid\mid ^{2}_{F} \mid\mid B\mid\mid ^{2}_{F} $$
So far I evaluated :
$$ \mid\mid \tilde{A}B - AB \mid\mid^{2}_{F} = \begin{cases} \sum_{ij} \sum_k(1-\frac{1}{p_{ij}})\mid a_{ik}b_{kj}\mid^2, & \text{with prob. $p_{ij}$} \\ \sum_{ij} \sum_k\mid a_{ik}b_{kj}\mid^2, & \text{with prob. $1 - p_{ij}$} \end{cases}$$
I am unsure how to take the expectation value of this, and I'm guessing to use the Cauchy-Schwartz inequality too.