Scalable Convex Methods for Phase Retrieval

This paper describes scalable convex optimization methods for phase retrieval. The main characteristics of these methods are the cheap per-iteration complexity and the low-memory footprint. With a variant of the original PhaseLift formulation, we first illustrate how to leverage the scalable Frank-Wolfe (FW) method (also known as the conditional gradient algorithm), which requires a tuning parameter. We demonstrate that we can estimate the tuning parameter of the FW algorithm directly from the measurements, with rigorous theoretical guarantees. We then illustrate numerically that recent advances in universal primal-dual convex optimization methods offer significant scalability improvements over the FW method, by recovering full HD resolution color images from their quadratic measurements.