Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval

TL;DR

This paper introduces IncrePR, a scalable incremental nonconvex optimization algorithm for phase retrieval that improves computational efficiency and recovery accuracy over existing methods, especially for structured measurements.

Contribution

Proposes IncrePR, a novel incremental nonconvex approach that relaxes the SDP formulation for phase retrieval, reducing computational cost and eliminating the need for good initialization.

Findings

Outperforms state-of-the-art nonconvex solvers in Gaussian measurement scenarios.

Achieves sharper phase transition for perfect recovery.

Effective for structured non-Gaussian measurements with multiple restarts.

Abstract

We aim to find a solution $\bm{x}\in\mathbb{C}^n$ to a system of quadratic equations of the form $b_i=\lvert\bm{a}_i^*\bm{x}\rvert^2$, $i=1,2,\ldots,m$, e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as \texttt{IncrePR}, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive computational cost of typical SDP solvers as well as the need of a good initialization for typical nonconvex methods. For Gaussian measurements, which is usually needed for provable convergence of nonconvex methods, \texttt{IncrePR} with restart strategy outperforms state-of-the-art nonconvex solvers with a sharper phase transition of perfect recovery and typical convex solvers in terms of computational cost and storage. For more challenging structured (non-Gaussian) measurements often occurred in real applications, such as transmission matrix and oversampling Fourier transform, \texttt{IncrePR} with several restarts can be used to find a good initial guess. With further refinement by local nonconvex solvers, one can achieve a better solution than that obtained by applying nonconvex solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.