# Optimal linear measurement operator

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, Can I really double my accuracy? On variance of a sum of random variables which is a (very) simple special case of the above.

Edit: $m,n$ are fixed. Edit 2: I want to find $f:R^m\to R^n$ and a linear $A:R^n\to R^m$ so as to minimize $$\sup_{||x||_2\le 1} E||x-f(Ax+\xi)||^2_2.$$

• The question is w.r.t. a fixed value of m? – Mark L. Stone Apr 29 '18 at 18:08
• Yes, fixed -- otherwise I can make $E||x-\hat x||^2$ arbitrarily small. I'll update. – Aryeh Kontorovich Apr 29 '18 at 18:37
• You want to minimise $\sup_{x\in R^n}E\|x-\hat{x}\|_2^2$ , with respect to $A$ and $f$, right? (I understand you have a function-valued minimisation problem, which for me needs some clarification, perhaps different from my guess. Else, $f$ identically to $x$ would trivially solve the problem.) – Lutz Mattner Apr 30 '18 at 9:57
• Yes, I've added a further clarification. – Aryeh Kontorovich Apr 30 '18 at 10:29
• This sounds a little bit like "optimal experimental design". – Dirk May 1 '18 at 15:02