  In many areas of imaging science, it is difficult to measure the phase of
linear measurements. As such, one often wishes to reconstruct a signal from
intensity measurements, that is, perform phase retrieval. In several
applications the signal in question is believed to be sparse. In this paper, we
use ideas from the recently developed polarization method for phase retrieval
and provide an algorithm that is guaranteed to recover a sparse signal from a
number of phaseless linear measurements that scales linearly with the sparsity
of the signal (up to logarithmic factors). This is particularly remarkable
since it is known that a certain popular class of convex methods is not able to
perform recovery unless the number of measurements scales with the square of
the sparsity of the signal. This is a shorter version of a more complete
publication that will appear elsewhere.
