Deterministic fast and stable phase retrieval in multiple dimensions

We present the first phase retrieval algorithm guaranteed to solve the multidimensional phase retrieval problem in polynomial arithmetic complexity without prior information. The method successfully terminates in O(N log(N)) operations for Fourier measurements with cardinality N. The algorithm is guaranteed to succeed for a large class of objects, which we term "Schwarz objects". We further present an easy-to-calculate and well-conditioned diagonal operator that transforms any feasible phase-retrieval instance into one that is solved by our method. We derive our method by combining techniques from classical complex analysis, algebraic topology, and modern numerical analysis. Concretely, we pose the phase retrieval problem as a multiplicative Cousin problem, construct an approximate solution using a modified integral used for the Schwarz problem, and refine the approximate solution to an exact solution via standard optimization methods. We present numerical experimentation demonstrating our algorithm's performance and its superiority to existing method. Finally, we demonstrate that our method is robust against Gaussian noise.