Phase retrieval with the alternating projections method

Phase retrieval problems consist in recovering unknown vectors with complex coordinates from the modulus of linear measurements. Despite being NP-hard in the worst case, these problems can be solved, with high probability, with simple and computationally efficient non-convex heuristics when the measurement vectors are random (typically realizations of a normal law). I will describe the most classical of these heuristics and the recovery guarantees that we can establish for it. I will also discuss the main questions that remain to be answered in order to fully understand its numerical behavior.