One-bit compressed sensing with partial Gaussian circulant matrices

Abstract

AbstractIn 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 $\delta$, $m\simeq \delta ^{-4} s\log (N/s\delta )$ 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 $\ell _1/\ell _2$ restricted isometry property property. Under a slightly worse dependence on $\delta$, we establish stability with respect to approximate sparsity, as well as full vector recovery results, i.e., estimation of both vector norm and direction.