Derandomized compressed sensing with nonuniform guarantees for $\ell_1$ recovery

Abstract: We extend the techniques of H\"{u}gel, Rauhut and Strohmer (Found. Comput. Math., 2014) to show that for every $\delta\in(0,1]$, there exists an explicit random $m\times N$ partial Fourier matrix $A$ with $m=s\operatorname{polylog}(N/\epsilon)$ and entropy $s^\delta\operatorname{polylog}(N/\epsilon)$ such that for every $s$-sparse signal $x\in\mathbb{C}^N$, there exists an event of probability at least $1-\epsilon$ over which $x$ is the unique minimizer of $|z|_1$ subject to $Az=Ax$. The bulk of our analysis uses tools from decoupling to estimate the extreme singular values of the submatrix of $A$ whose columns correspond to the support of $x$.