Alternating projections for phase retrieval with random sensing vectors

Phase retrieval problems are a subclass of low-rank matrix recovery problems, that have been studied for a long time because of their important applications in physics. They have traditionally been solved with non-convex algorithms, that came with no theoretical convergence guarantees, and were known to sometimes get stuck in local optima. In the past few years, new algorithms have been proposed, that provably reconstruct the true solution with high probability for random instances of phase retrieval problems. We will show that, in this "random" setting, the most well-known traditional algorithm actually enjoys essentially the same convergence guarantees as the newer methods, provided that it is preceded by a suitable initialization procedure. We will discuss the importance of the initialization.