Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5773600 | Applied and Computational Harmonic Analysis | 2017 | 13 Pages |
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 â1 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Paul Hand,