  Consider the task of recovering an unknown $n$-vector from phaseless linear
measurements. This task is the phase retrieval problem. Through the technique
of lifting, this nonconvex problem may be convexified into a semidefinite
rank-one matrix recovery problem, known as PhaseLift. Under a linear number of
exact Gaussian measurements, PhaseLift recovers the unknown vector exactly with
high probability. Under noisy measurements, the solution to a variant of
PhaseLift has error proportional to the $\ell_1$ norm of the noise. In the
present paper, we study the robustness of this variant of PhaseLift to a case
with noise and gross, arbitrary corruptions. We prove that PhaseLift can
tolerate a small, fixed fraction of gross errors, even in the highly
underdetermined regime where there are only $O(n)$ measurements. The lifted
phase retrieval problem can be viewed as a rank-one robust Principal Component
Analysis (PCA) problem under generic rank-one measurements. From this
perspective, the proposed convex program is simpler that the semidefinite
version of the sparse-plus-low-rank formulation standard in the robust PCA
literature. Specifically, the rank penalization through a trace term is
unnecessary, and the resulting optimization program has no parameters that need
to be chosen. The present work also achieves the information theoretically
optimal scaling of $O(n)$ measurements without the additional logarithmic
factors that appear in existing general robust PCA results.
