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Motivated by the research on sampling problems for a union of subspaces(UoS), we investigate in this paper the phase-retrieval problem for the signalsthat are residing in a union of (finitely generated) cones (UoC for short) in$\mathbb{R}^{n}$. We propose a two-step PR-scheme:$\hbox{PR}=\hbox{detection}+\hbox{recovery}$. We first establish a sufficientand necessary condition for the detectability of a UoC, and then design adetection algorithm that allows us to determine the cone where the targetsignal is residing. The phase-retrieval will be then performed within thedetected cone, which can be achieved by using at most $\Gamma$-number ofmeasurements and with very low complexity, where $\Gamma (\leq n)$ is themaximum of the ranks of the generators for the UoC. Numerical experiments areprovided to demonstrate the efficiency of our approach, and to exhibitcomparisons with some existing phase-retrieval methods.

Two-step PR-scheme for recovering signals in detectable union of cones by magnitude measurements