# On Fienup Methods for Regularized Phase Retrieval

## TL;DR

This paper provides new theoretical insights into Fienup methods for phase retrieval, showing their convergence properties and proposing an accelerated version that performs well with sparse signals.

## Contribution

It offers a novel analysis of Fienup methods as regularized nonconvex optimization and introduces an accelerated algorithm for sparse phase retrieval.

## Key findings

- Fienup methods can be viewed as alternating minimization on a regularized nonconvex problem.
- Under semi-algebraic priors, the algorithm converges smoothly to a critical point.
- The proposed accelerated method with an $ ext{l}_1$ prior outperforms existing approaches in sparse 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.