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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 explicitrandom $m\times N$ partial Fourier matrix $A$ with$m=s\operatorname{polylog}(N/\epsilon)$ and entropy$s^\delta\operatorname{polylog}(N/\epsilon)$ such that for every $s$-sparsesignal $x\in\mathbb{C}^N$, 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 theextreme singular values of the submatrix of $A$ whose columns correspond to thesupport of $x$.

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