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

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.02529
 Bibcode:
 2020arXiv200102529J
 Keywords:

 Computer Science  Information Theory
 EPrint:
 19 pages, 2 figures. (Note: the previous small side result on perfect signal recovery from modulo pi phases was wrong