Phase Retrieval via Wirtinger Flow: Theory and Algorithms

TLDR

Phase retrieval aims to reconstruct a complex signal from magnitude‑only measurements, a problem that is ill‑posed without phase information and whose analysis offers insights into non‑convex optimization schemes with broader computational implications. This work formulates phase retrieval as a non‑convex optimization problem and proposes a concrete algorithm to solve it. The algorithm begins with a spectral initialization and then iteratively refines the estimate using low‑complexity update rules akin to gradient descent, and its effectiveness is demonstrated on image data. The method provably recovers the signal from a nearly minimal number of random measurements, converges geometrically, is computationally efficient, and can be extended to a near‑linear time algorithm for coded diffraction patterns.

Abstract

We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x of C^n about which we have phaseless samples of the form y_r = |< a_r,x >|^2, r = 1,2,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a non-convex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of non-convex optimization schemes that may have implications for computational problems beyond phase retrieval.