FROG, Nonlinear Compressive Sensing ? continued.

After reading the entry on whether FROG is an instance of Nonlinear Compressed Sensing ?, Rick Trebino wrote the following in the comment section:

Recent Georgia Tech (Ph.D.) grad, John Bowlan, pointed me to your site. Thanks, Igor, for the nice summary of my work (and unsolicited plug for my book)! I'd be interested in exploring any links between FROG and your field. I'll be meeting with Justin Romberg, and we'll see what emerges. Also, I'd be happy to hear from any other of your readers (rick.trebino@physics.gatech.edu). And, by the way, we're still waiting for that proof; any ideas?

To summarize, Rick has devised a reconstruction algorithm based on analog measurements implemented on the GRENOUILLE hardware. The reconstruction algorithm is pictured below.

Even though, the measurement is nonlinear, I suspect the reconstruction works because the solution is sparse (devising the shape of the smallest pusle possible). The sets on which the projections are made are not convex. The whole analysis is reminiscent of the approach of Stefano Marchesini on Ab initio Phase Retrieval Algorithms. as well as the framework of proximal splitting ( see Proximal Splitting Methods in Signal Processing by Patrick L. Combettes, and Jean-Christophe Pesquet )

and the other references mentioned earlier in FROG is an instance of Nonlinear Compressed Sensing ?.