On the Information-Theoretic Limits of Noisy Sparse Phase Retrieval

The support recovery problem consists of determining a sparse subset of variables that is relevant in generating a set of observations. In this paper, we study the support recovery problem in the phase retrieval model consisting of noisy phaseless measurements, which arises in a diverse range of settings such as optical detection, X-ray crystallography, electron microscopy, and coherent diffractive imaging. Our focus is on information- theoretic fundamental limits under an approximate recovery criterion, with Gaussian measurements and a simple discrete model for the sparse non-zero entries. Our bounds provide sharp thresholds with near-matching constant factors in several scaling regimes on the sparsity and signal-to-noise ratio.