Sparse Phase Retrieval with Partial Convolutional Measurements

In this paper, we consider the sparse phase retrieval problem, that is, recovering an unknown s-sparse signal x ∈ R ⁿ from intensity-only measurements. Specifically, we focus on the problem of recovering x from the observations that are cyclically convoluted with a kernel a ∈ R ⁿ , denoted as y = | R_I ( a ⊛ x )| ² . Here, R_I : R ⁿ → R ^|I| represents a random subsampled operator that is restricted to the index set I . This model is motivated by real-world applications in optics and communications. We provide that if a is a random Gaussian vector and the number of subsampled measurements is on the order of spolylog( n ), one can recover x up to a global phase provided that the initialization estimator is around x . It is the first theoretical result that discusses the behavior of sparse convolutional phase retrieval under physically realistic measurements, as opposed to independent Gaussian measurements.