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

TL;DR

This paper introduces a convex programming approach called PhaseLift for exact and stable recovery of signals from magnitude-only measurements, requiring roughly n log n measurements and robust to noise.

Contribution

It proves that phase retrieval can be solved exactly using semidefinite programming with random measurements, a significant advancement over previous non-convex methods.

Findings

Exact recovery with high probability using O(n log n) measurements

Robustness of the method in the presence of noise

Demonstration that convex programming can solve phase retrieval efficiently

Abstract

Suppose we wish to recover a signal x in C^n from m intensity measurements of the form |<x,z_i>|^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors 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 a vis additive noise.