Training neural networks with iterative projection algorithms

Previously, I was working as a physicist in the area of phase retrieval (PR). PR is concerned with finding the phase of a complex valued function (typically in Fourier space) given knowledge of its amplitude, along with constraints in real-space (things like positivity and finite extent).

PR is a non-convex optimization problem and has been the subject of quite a lot of work and forms the backbone of crystallography, a stalwart of structural biology.

Some of the most successful algorithms for the general PR problem are projection based methods, inspired by convex optimizations projection onto convex sets (for an excellent overview see [Marchesini2007]). Due to the projection based methods success in PR, I wondered if it would be possible to train a neural net using something similar.