The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold
This solves open problems in phase retrieval, offering theoretical guarantees for signal reconstruction, but it is incremental as it builds on existing algebraic approaches.
The paper tackles the phase retrieval problem from magnitude measurements by treating it as an algebraic estimation problem, showing that a certain number of generic measurements enable signal reconstruction for generic or all signals, and provides a closed-form algebraic technique with non-asymptotic guarantees.
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.