Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation

TL;DR

This paper introduces a new convex relaxation method for phase retrieval that operates directly in the signal domain, avoiding costly lifting procedures and providing theoretical guarantees with high probability for random measurements.

Contribution

It presents a flexible convex relaxation approach that is computationally efficient and backed by geometric and statistical learning theory analysis, outperforming traditional SDP methods.

Findings

High probability success with random measurements

Effective phase transition behavior demonstrated in simulations

Applicable to coded diffraction measurement scenarios

Abstract

We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with "lifting" and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric "slab". Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector. We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. Phase transition of our estimator is evaluated through simulations. Our numerical experiments also suggest that the proposed method can solve phase retrieval problems with coded diffraction measurements as well.