Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval

Abstract

We aim to find a solution ${\varvec{x}}\in {\mathbb {C}}^n$ to a system of quadratic equations of the form $b_i=|{\varvec{a}}_i^*{\varvec{x}}|^2$, $i=1,2,\ldots ,m$, 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 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, restart-IncrePR solving three consecutive PhaseLift problems outperforms state-of-the-art nonconvex gradient flow based 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, IncrePR with several consecutive repeats 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 gradient flow based solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.