This is problem 10.3.6a in Vershynin's *High Dimensional Probability* book that I'm self-studying. I wasn't sure how to do this problem. I tried finding a bound on the set the solution comes from and using theorem 10.3.4 in the book, but I wasn't able to do it.

Assume $A$ is an $m$ by $n$ matrix with independent, isotropic, and subgaussian rows $A_i$. Suppose $y$ is a know measurement and $Ax=y$. Show that an unknown $s$-sparse signal $x$ (i.e. at most $s$ nonzero entries) can be approximately recovered by solving the convex optimization problem

$$\text{minimize } \|x'\|_1 \text{ s.t. } y=Ax'$$

The recovery error satisfies

$$\mathbb{E} \|\hat{x}-x\|_2 \leq C\sqrt{\frac{s\log n}{m}} \|x\|_2 $$

where $\hat{x}=\text{argmin}_{Ax'=y}||x'||_1$