PhaseLift is robust to a constant fraction of arbitrary errors

Abstract

Abstract Consider the task of recovering an unknown n-vector from phaseless linear measurements. This nonconvex problem may be convexified into a semidefinite rank-one matrix recovery problem, known as PhaseLift. Under a linear number of 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 l 1 norm of the noise. In the present paper, we study the robustness of this variant of PhaseLift to gross, arbitrary corruptions. We prove that PhaseLift can tolerate noise and 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 than 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.