  We study a class of real robust phase retrieval problems under a Gaussian
assumption on the coding matrix when the received signal is sparsely corrupted
by noise. The goal is to establish conditions on the sparsity under which the
input vector can be exactly recovered. The recovery problem is formulated as
the minimization of the $\ell_1$ norm of the residual.
  The main contribution is a robust phase retrieval counterpart to the seminal
paper by Candes and Tao on compressed sensing ($\ell_1$ regression) [Decoding
by linear programming. IEEE Transactions on Information Theory,
51(12):4203-4215, 2005]. Our analysis depends on a key new property on the
coding matrix which we call the {Absolute Range Property} (ARP). This property
is an analogue to the Null Space Property (NSP) in compressed sensing. When the
residuals are computed using squared magnitudes, we show that ARP follows from
a standard Restricted Isometry Property (RIP). However, when the residuals are
computed using absolute magnitudes, a new and very different kind of RIP or
growth property is required. We conclude by showing that the robust phase
retrieval objectives are sharp with respect to their minimizers with high
probability.
