An Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax

TL;DR

This paper provides a simple, elementary proof that the PhaseMax convex program can exactly recover real-valued signals in phase retrieval using minimal measurements, avoiding complex theoretical tools.

Contribution

It introduces a straightforward proof for PhaseMax's effectiveness in real-valued phase retrieval, simplifying previous complex proofs and relying on basic probabilistic arguments.

Findings

PhaseMax achieves exact recovery with optimal sample complexity.

The proof uses standard concentration and covering arguments.

The approach simplifies understanding of phase retrieval guarantees.

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. Very 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 current 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.