PhaseLift is robust to a constant fraction of arbitrary errors

Consider the task of recovering an unknown n-vector from phaseless linear measurements. This task is the phase retrieval problem. Through the technique of lifting, this nonconvex problem may be convexified into a semidefinite rank-one matrix recovery problem, known as PhaseLift. Under a linear number of exact 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 ℓ1 norm of the noise. In the present paper, we study the robustness of this variant of PhaseLift to a case with noise and gross, arbitrary corruptions. We prove that PhaseLift can tolerate 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 that 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. The present work also achieves the information theoretically optimal scaling of O(n) measurements without the additional logarithmic factors that appear in existing general robust PCA results.