Let $x\in R^n$ be an unknown vector. Suppose I am allowed to choose any $A\in R^{m\times n}$, under the constraint that each row of $A$ has $\ell_2$ norm at most $1$. Then I carry out a "measurement", which means that I get to observe $Ax+\xi$, where $y=\xi\sim N(0,I_m)$ is iid noise. Finally, I am allowed to apply an arbitrary deterministic function $f:R^m\to R^n$ to estimate $x$ via
$\hat x = f(y)$. Question: under the specified constraints, which choice of $A$ (and $f$) minimizes $E||x-\hat x||^2_2$? Has this class of problems been studied somewhere? I was inspired by this question,
https://mathoverflow.net/questions/297755/can-i-really-double-my-accuracy-on-variance-of-a-sum-of-random-variables
which is a (very) simple special case of the above.