On the Global Minimizers of Real Robust Phase Retrieval with Sparse Noise

We study a class of real robust phase retrieval problems under a Gaussian assumption on the coding matrix when the received signal is sparsely corrupted by noise. The goal is to establish conditions on the sparsity under which the input vector can be exactly recovered. The recovery problem is formulated as the minimization of the $\ell_1$ norm of the residual. The main contribution is a robust phase retrieval counterpart to the seminal paper by Candes and Tao on compressed sensing ($\ell_1$ regression) [Decoding by linear programming. IEEE Transactions on Information Theory, 51(12):4203-4215, 2005]. Our analysis depends on a key new property on the coding matrix which we call the {Absolute Range Property} (ARP). This property is an analogue to the Null Space Property (NSP) in compressed sensing. When the residuals are computed using squared magnitudes, we show that ARP follows from a standard Restricted Isometry Property (RIP). However, when the residuals are computed using absolute magnitudes, a new and very different kind of RIP or growth property is required. We conclude by showing that the robust phase retrieval objectives are sharp with respect to their minimizers with high probability.