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 (4)

• Robust Principal Component Analysis?
• 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
• Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow

Cited by in corpus (8)

• Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation
• Robust Wirtinger Flow for Phase Retrieval with Arbitrary Corruption
• BranchHull: Convex bilinear inversion from the entrywise product of signals with known signs
• Solving Equations of Random Convex Functions via Anchored Regression
• Fundamental Limits of PhaseMax for Phase Retrieval: A Replica Analysis
• Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces
• Phase retrieval of complex-valued objects via a randomized Kaczmarz method
• Phase Retrieval by Alternating Minimization with Random Initialization