  The problem of reconstructing a sparse signal vector from magnitude-only
measurements (a.k.a., compressive phase retrieval), emerges naturally in
diverse applications, but it is NP-hard in general. Building on recent advances
in nonconvex optimization, this paper puts forth a new algorithm that is termed
compressive reweighted amplitude flow and abbreviated as CRAF, for compressive
phase retrieval. Specifically, CRAF operates in two stages. The first stage
seeks a sparse initial guess via a new spectral procedure. In the second stage,
CRAF implements a few hard thresholding based iterations using reweighted
gradients. When there are sufficient measurements, CRAF provably recovers the
underlying signal vector exactly with high probability under suitable
conditions. Moreover, its sample complexity coincides with that of the
state-of-the-art procedures. Finally, substantial simulated tests showcase
remarkable performance of the new spectral initialization, as well as improved
exact recovery relative to competing alternatives.
