Sparse phase retrieval via iteratively reweighted amplitude flow

Sparse phase retrieval (PR) aims at reconstructing a sparse signal vector from a few phaseless linear measurements. It emerges naturally in diverse applications, but it is NP-hard in general. Drawing from advances in nonconvex optimization, this paper presents a new algorithm that is termed compressive reweighted amplitude flow (CRAF) for sparse PR. CRAF operates in two stages: Stage one computes an initial guess by means of a new spectral procedure, and stage two implements a few hard thresholding based iteratively reweighted gradient iterations on the amplitude-based least-squares cost. When there are sufficient measurements, CRAF reconstructs the true signal vector exactly under suitable conditions. Furthermore, its sample complexity coincides with that of the state-of-the-art approaches. Numerical experiments showcase improved performance of the proposed approach relative to existing alternatives.