Corruption Robust Phase Retrieval via Linear Programming

Abstract

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 $|⟨a_i,x_0⟩|+{\eta }_i,i=1,2\dots m$ 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.

References in corpus (2)

• Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation
• An Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax

Cited by in corpus (1)

• Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation