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# Communications in Mathematical Sciences

## Volume 16 (2018)

### Number 7

### An elementary proof of convex phase retrieval in the natural parameter space via the linear program PhaseMax

Pages: 2047 – 2051

(Fast Communication)

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n7.a13

#### Authors

#### Abstract

The phase retrieval problem has garnered significant attention since the development of the PhaseLift algorithm, which is a convex program that operates in a lifted space of matrices. Because of the substantial computational cost due to lifting, many approaches to phase retrieval have been developed, including non-convex optimization algorithms which operate in the natural parameter space, such as Wirtinger Flow. Recently, a convex formulation called PhaseMax has been discovered, and it has been proven to achieve phase retrieval via linear programming in the natural parameter space under optimal sample complexity. The original proofs of PhaseMax rely on statistical learning theory or geometric probability theory. Here, we present a short and elementary proof that PhaseMax exactly recovers real-valued vectors from random measurements under optimal sample complexity. Our proof only relies on standard probabilistic concentration and covering arguments, yielding a simpler and more direct proof than those that require statistical learning theory, geometric probability or the highly technical arguments for Wirtinger Flow-like approaches.

#### Keywords

phase retrieval, PhaseMax, linear programming

#### 2010 Mathematics Subject Classification

60D05, 90C05, 94A12

Received 4 March 2018

Accepted 15 September 2018

Published 7 March 2019