  We consider the problem of phase retrieval from corrupted magnitude
observations. In particular we show that a fixed $x_0 \in \mathbb{R}^n$ can be
recovered exactly from corrupted magnitude measurements $|\langle a_i, x_0
\rangle | + \eta_i, \quad i =1,2\ldots m$ with high probability for $m = O(n)$,
where $a_i \in \mathbb{R}^n$ are i.i.d standard Gaussian and $\eta \in
\mathbb{R}^m$ has fixed sparse support and is otherwise arbitrary, by using a
version of the PhaseMax algorithm augmented with slack variables subject to a
penalty. This linear programming formulation, which we call RobustPhaseMax,
operates in the natural parameter space, and our proofs rely on a direct
analysis of the optimality conditions using concentration inequalities.
