Title: On Fienup Methods for Regularized Phase Retrieval,
Abstract: Alternating minimization, or Fienup methods, have a long history in phase
retrieval. We provide new insights related to the empirical and theoretical
analysis of these algorithms when used with Fourier measurements and combined
with convex priors. In particular, we show that Fienup methods can be viewed as
performing alternating minimization on a regularized nonconvex least-squares
problem with respect to amplitude measurements. We then prove that under mild
additional structural assumptions on the prior (semi-algebraicity), the
sequence of signal estimates has a smooth convergent behaviour towards a
critical point of the nonconvex regularized least-squares objective. Finally,
we propose an extension to Fienup techniques, based on a projected gradient
descent interpretation and acceleration using inertial terms. We demonstrate
experimentally that this modification combined with an $\ell_1$ prior
constitutes a competitive approach for sparse phase retrieval.