An elementary proof of a fundamental result in phase retrieval

TL;DR

This paper provides an elementary proof of a key phase retrieval theorem in real space, simplifying previous algebraic geometry-based proofs and extending the analysis to complex spaces, with applications to norm retrieval classifications.

Contribution

It offers an accessible proof of Edidin's phase retrieval theorem in real space and extends the analysis to complex space, addressing the theorem's limitations and applications.

Findings

Elementary proof of phase retrieval in real space

Complex space version shows 'if' part holds, 'only if' fails

Verification of classifications of norm retrieval

Abstract

Edidin [3] proved a fundamental result in phase retrieval: Theorem: A family of orthogonal projections $\{P_i\}_{i=1}^m$ does phase retrieval in $\mathbb{R}^n$ if and only if for every $0\not= x\in \mathbb{R}^n$, the family $\{P_ix\}_{i=1}^m$ spans $\mathbb{R}^n$. The proof of this result relies on Algebraic Geometry and so is inaccessible to many people in the field. We will give an elementary proof of this result without Algebraic Geometry. We will also solve the complex version of this result by showing that the "if" part fails and the "only if" part holds in $\mathbb{C}^n$. Finally, we will show that these techniques can be used to verify two classifications of norm retrieval.