# Problem 24: “Ultimate” Deterministic Sparse Recovery

We say that a vector $v \in \mathbb R^n$ is $k$-sparse for some $k \in \{0,\ldots,n\}$ if there are no more than $k$ non-zero coordinates in $v$. The goal in the problem being considered is to design an $m \times n$ matrix $A$ such that for any $x \in R^n$, one can recover from $Ax$ a vector $x^* \in \mathbb R^n$ that satisfies the following “$L_2/L_1$” approximation guarantee:$\left\|x^*-x\right\|_2 \leq \min_{k\text{-sparse}\,x'\in\mathbb R^n} \frac{C}{\sqrt{k}} \left\|x'-x\right\|_1,$where $C>0$ is a constant.
We conjecture that there is a solution that (a) uses $m=O(k \log (n/k))$ and (b) supports recovery algorithms running in time $O(n \operatorname{polylog} n)$.
It is known that one can achieve either (a) or (b) (see, e.g., [NeedellT-10]). It is also possible to achieve both (a) and (b), but with a different “$L_1/L_1$” approximation guarantee, where $\|x^*-x\|_1 \leq \min_{k\text{-sparse}\,x'} C \|x'-x\|_1$ [IndykR-08,BerindeIR-08].