The semidefinite program PhaseLift can solve phase retrieval with corrupted data.
Paul Hand, Rice University

Phase retrieval is the process of recovering a vector from phaseless linear measurements.  It is a challenging mathematical task that appears in X-ray crystallography and other applications.  The problem is nonconvex, and it can be convexified into a well-known semidefinite rank-one matrix recovery problem known as PhaseLift.  In some cases, the measurements can be corrupted.  This can happen, for example, when there are occlusions, sensor failures, or sensor saturation.  In this talk, we show that a variant of PhaseLift can successfully tolerate corrupted data.  Specifically, we show that under a Gaussian measurement model, any signal can be recovered with high probability, provided there are enough measurements and provided that at most a fixed fraction of the measurements are arbitrarily corrupted.