  We demonstrate that a sparse signal can be estimated from the phase of
complex random measurements, in a "phase-only compressive sensing" (PO-CS)
scenario. With high probability and up to a global unknown amplitude, we can
perfectly recover such a signal if the sensing matrix is a complex Gaussian
random matrix and the number of measurements is large compared to the signal
sparsity. Our approach consists in recasting the (non-linear) PO-CS scheme as a
linear compressive sensing model. We built it from a signal normalization
constraint and a phase-consistency constraint. Practically, we achieve stable
and robust signal direction estimation from the basis pursuit denoising
program. Numerically, robust signal direction estimation is reached at about
twice the number of measurements needed for signal recovery in compressive
sensing.
