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The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold

Király, FJ; Ehler, M; (2014) The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold. arXiv.org , Article arXiv:1402.4053 [math.FA]. Green open access

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Abstract

We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve a few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved by a closed-form algebraic estimation technique, termed ideal regression, providing non-asymptotic success guarantees.

Type: Article
Title: The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold
Open access status: An open access version is available from UCL Discovery
Publisher version: https://arxiv.org/abs/1402.4053
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Functional Analysis; Computer Vision and Pattern Recognition; Information Theory; Algebraic Geometry; Machine Learning
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/1517413
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