Corruption Robust Phase Retrieval via Linear Programming

We consider the problem of phase retrieval from corrupted magnitude observations. In particular we show that a fixed x0 ∈ Rn can be recovered exactly from corrupted magnitude measurements |〈ai, x0〉|+ ηi, i = 1, 2 . . . m with high probability for m = O(n), where ai ∈ Rn are i.i.d standard Gaussian and η ∈ Rm 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.