PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
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TLDR
It is shown that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques, and it is proved that the methodology is robust vis‐à‐vis additive noise.Abstract:
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 , ; that is, from data in which phase information is missing. We prove that if the vectors 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 , 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-a-vis additive noise. © 2012 Wiley Periodicals, Inc.read more
Citations
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Journal ArticleDOI
Phase Retrieval via Wirtinger Flow: Theory and Algorithms
TL;DR: In this article, a nonconvex formulation of the phase retrieval problem was proposed and a concrete solution algorithm was presented. But the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements.
Journal ArticleDOI
Phase Retrieval with Application to Optical Imaging: A contemporary overview
TL;DR: The goal is to describe the current state of the art in this area, identify challenges, and suggest future directions and areas where signal processing methods can have a large impact on optical imaging and on the world of imaging at large.
Book
High-Dimensional Statistics: A Non-Asymptotic Viewpoint
TL;DR: This book provides a self-contained introduction to the area of high-dimensional statistics, aimed at the first-year graduate level, and includes chapters that are focused on core methodology and theory - including tail bounds, concentration inequalities, uniform laws and empirical process, and random matrices.
Journal ArticleDOI
Phase Retrieval via Matrix Completion
TL;DR: This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging, and many other applications, and combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements.
Journal ArticleDOI
Phase recovery, MaxCut and complex semidefinite programming
TL;DR: In this article, the phase retrieval problem is cast as a nonconvex quadratic program over a complex phase vector and formulated a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program.
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