The Importance of Phase in Complex Compressive Sensing

We consider the question of estimating a real low-complexity signal (such as sparse vector or low-rank matrix) from phase complex random measurements. show that in this phase-only compressive sensing (PO-CS) scenario, we can perfectly recover such with high probability and up to global unknown amplitude if matrix is Gaussian number measurements large compared complexity level space. Our approach proceeds by recasting (non-linear) PO-CS scheme linear model built normalization constraint, phase-consistency constraint imposing any estimate match observed phases measurement domain. Practically, stable robust estimation direction achieved instance optimal algorithm literature basis pursuit denoising). This ensured proving associated equivalent satisfies restricted isometry property under above condition on finally observe experimentally recovery reached at about twice needed for sensing.