Algebraic Reconstruction Bounds and Explicit Inversion for Phase Retrieval at the Identifiability Threshold

We study phase retrieval from rank-one magnitude and more general linear magnitude measurements of an unknown signal as an algebraic estimation problem. 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 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.