The Gauss--Newton Method for Phase Retrieval from Coded Diffraction Patterns

We study the minimum-norm Gauss--Newton method for phase retrieval from random coded diffraction patterns with additive intensity noise. For bounded symmetric masks satisfying the standard complex moment conditions, we establish convergence guarantees with quadratic error recursions in both the resampled and fixed-data settings. In the resampled setting, $O(\log^2 n)$ masks per independent block suffice to obtain a spectral initializer within a constant-relative-radius neighborhood and to establish a local quadratic error recursion with an additive noise term. Consequently, in the noiseless case, the method achieves relative accuracy $ε$ using $O(\log\log(1/ε)\log^2 n)$ masks. In the fixed-data setting, $O(\log^6(en))$ masks suffice for spectral initialization within a neighborhood whose relative radius is proportional to $\log^{-5/2}(nL)$. Within this neighborhood, the method satisfies a linear contraction bound and eventually a quadratic error recursion. In particular, in the noiseless case, the method initially converges linearly and enters a quadratic convergence phase after $O(\log n)$ iterations, without acquiring additional masks. Numerical experiments on synthetic signals and a natural image demonstrate the method's efficiency relative to state-of-the-art algorithms.