PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming



Suppose we wish to recover a signal \input amssym $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} {\bi x} \in {\Bbb C}^n$ from m intensity measurements of the form equation image, equation image; that is, from data in which phase information is missing. We prove that if the vectors equation image are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program–-a trace-norm minimization problem; this holds with large probability provided that m is on the order of equation image, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis-à-vis additive noise. © 2012 Wiley Periodicals, Inc.