One-bit compressed sensing with partial Gaussian circulant matrices

In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $\delta$, $m\sim {\delta }^{-4}s\log (N\mathrm{/}s\delta )$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $\delta$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an ${\mathrm{\ell }}_1\mathrm{/}{\mathrm{\ell }}_2$ RIP-property. Under a slightly worse dependence on $\delta$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.