Derandomized compressed sensing with nonuniform guarantees for ${\ell }_1$ recovery

Abstract

We extend the techniques of Hügel, 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 $m=s\mathrm{polylog}(N/\epsilon )$ and entropy $s^{\delta }\mathrm{polylog}(N/\epsilon )$ 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 $\Vert z{\Vert }_1$ 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 $x$.