Phase retrieval via linear programming: Fundamental limits and algorithmic improvements

Abstract

A recently proposed convex formulation of the phase retrieval problem estimates the unknown signal by solving a simple linear program. This new scheme, known as PhaseMax, is computationally efficient compared to standard convex relaxation methods based on lifting techniques. In this paper, we present an exact performance analysis of PhaseMax under Gaussian measurements in the large system limit. In contrast to previously known performance bounds in the literature, our results are asymptotically exact and they also reveal a sharp phase transition phenomenon. Furthermore, the geometrical insights gained from our analysis led us to a novel nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm based on successive linearization and maximization over a polytope. This new algorithm, which we call PhaseLamp, has provably superior recovery performance over the original PhaseMax method.

A Precise Analysis of PhaseMax in Phase Retrieval

“…where g ∈ R m and h ∈ R 2n−2 with entries drawn independently from standard normal distribution. Analysis of (20) is similar to [20]. Due to lack of space, we defer technical details to the full version of the paper.…”

“…We conclude the paper with a theorem that characterizes the performance of the ERO. Let w * be the optimizer of (20). Define s * := 1 + w * 1 and t * := ||w * ||.…”

“…The exact phase transition threshold, i.e. the exact value of the constant C, for the real PhaseMax has been recently derived in [20], [21]. However, for the practical case of complex signals, previous results could only provide an upper bound on C.…”

“…al. [20] to analyze the real version of the PhaseMax. But unfortunately, the complex case does not directly fit into the framework of CGMT.…”

PhaseLin: Linear phase retrieval

“…In Figure 1, we show the recovered images for a 16 × 16pixel image taken from the dataset provided in [28] with M = 3N and M = 9N measurements, respectively. We compare PhaseLin to the Wirtinger flow (WF) [14], reweighed amplitude flow (RAF) [22], Fienup [2], PhaseMax [23]- [25], PhaseLamp [26], Gerchberg-Saxton (GS) [1], and PhaseLift [9] [14] 0.61 0.4492 0.13 0.3069 RAF [22] 0.70 0.4769 0.16 0.2946 Fienup [2] 19.1 0.6070 0.45 0.2899 PhaseMax [25] 0.96 0.6254 0.42 0.4872 PhaseLamp [26] 13.3 0.6843 5.31 0.6848 GS [1] 17.9 0.6036 0.43 0.2899 PhaseLift [9] 170 0.3195 35.0 0.2786 methods. For each method, we use the asymptotically-optimal spectral initializer [20].…”