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

• Published in: Communications on pure and applied mathematics Vol. 66; no. 8; pp. 1241 - 1274
• Main Authors: Candès, Emmanuel J., Strohmer, Thomas, Voroninski, Vladislav
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.08.2013; John Wiley and Sons, Limited
• Subjects: Mathematical problems; Mathematical programming; Measurement; Probability distribution
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21432

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 $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} |\langle \bi x,\bi z_i \rangle|^2$, $i = 1, 2, \ldots, m$; that is, from data in which phase information is missing. We prove that if the vectors $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}}{\bi z}_i$ 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 $n {\log n}$, 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.