CS: Sparse Spectral Factorization for FROG ?

If you recall, when Rick Trebino has to determine the time width of the shortest time scale possible, he makes the case that an autocorrelation is not a good transform he can use to deconvolute that shortest time scale signal. He then goes on devising a hardware setup called FROG that allows him to perform this operation. A while ago, I asked the question as to whether FROG was not connected to compressed sensing in some sense ( Is FROG an instance of Nonlinear Compressed Sensing ? , FROG, Nonlinear Compressive Sensing ? continued., A small Q&A with Rick Trebino, the inventor of FROG. ). Well it looks like we have the beginning of an answer for the autocorrelation business, it looks like if the signal is sparse, then that operation seems to be valid. What about FROG ? here is the paper for the autocorrelation: Sparse Spectral Factorization: Unicity and Reconstruction Algorithms by Yue M. Lu and Martin Vetterli. The abstract reads:

Spectral factorization is a classical tool in signal processing and communications. It also plays a critical role in X-ray crystallography, in the context of phase retrieval. In this work, we study the problem of sparse spectral factorization, aiming to recover a one-dimensional sparse signal from its autocorrelation. We present a sufficient condition for the recovery to be unique, and propose an iterative algorithm that can obtain the original signal (up to a sign change, time-shift and time-reversal). Numerical simulations verify the effectiveness of the proposed algorithm