Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval

We develop procedures, based on minimization of the composition $f(x) = h(c(x))$ofaconvexfunction$h$andsmoothfunction$c$, for solving random collections of quadratic equalities, applying our methodology to phase retrieval problems. We show that the prox-linear algorithm we develop can solve phase retrieval problems—even with adversarially faulty measurements—with high probability as soon as the number of measurements $m$ is a constant factor larger than the dimension $n$ of the signal to be recovered. The algorithm requires essentially no tuning—it consists of solving a sequence of convex problems—and it is implementable without any particular assumptions on the measurements taken. We provide substantial experiments investigating our methods, indicating the practical effectiveness of the procedures and showing that they succeed with high probability as soon as $m\mathrm{/}n\ge 2$ when the signal is real-valued.