An elementary proof of a fundamental result in phase retrieval

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Casazza, Peter G, Tremain, Janet C
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 10.03.2021
Subjects
Algebra
Phase retrieval
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Summary: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.
ISSN:2331-8422