  We consider the question of estimating a real low-complexity signal (such as
a sparse vector or a low-rank matrix) from the phase of complex random
measurements. We show that in this "phase-only compressive sensing" (PO-CS)
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 (non-linear)
PO-CS scheme as a linear compressive sensing model built from a signal
normalization constraint, and a phase-consistency 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.
