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

Motivated by the research on sampling problems for a union of subspaces (UoS), we investigate in this paper the phase-retrieval problem for the signals that 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 sufficient and necessary condition for the detectability of a UoC, and then design a detection algorithm that allows us to determine the cone where the target signal is residing. The phase-retrieval will be then performed within the detected cone, which can be achieved by using at most $\Gamma$-number of measurements and with very low complexity, where $\Gamma (\leq n)$ is the maximum of the ranks of the generators for the UoC. Numerical experiments are provided to demonstrate the efficiency of our approach, and to exhibit comparisons with some existing phase-retrieval methods.