Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval

Abstract

We aim to find a solution x ∈ C n to a system of quadratic equations of the form b i = |a * i x| 2 , i = 1, 2, . . . , 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.

Solving phase retrieval via graph projection splitting

“…GPS shows the sharpest phase transition among all methods, successfully solving problem (1) with m = 1.7n and m = 2.7n measurements for real and complex cases respectively. This phase transition is also better than the incremental nonconvex approach for PhaseLift [13].…”

“…Hence the matrix B depends on the iteration number k, but we do not explicitly show the dependence to simplify the notation. The proximity term in (13) is nonlinear which has the following directional derivative 2 for z ∈ C n and h ∈ C n ,…”

“…We use the default configuration of the parameters described in the corresponding references. Note that here we only consider the nonconvex approaches for comparison, as the comparison of nonconvex solvers to SDR PhaseLift approach can be found in [13]. All tests are conducted with Monte-Carlo simulations.…”

“…Another convexification algorithm PhaseLin is introduced with the help of anchor vector [8], which plays the similar role as initialization in gradient flow algorithms. To reduce the high computation cost of PhaseLift, matrix-factorization based approach, IncrePR [13], with improved phase transition is developed. The main issue of these solvers is their inapplicability to tackle practical phase retrieval problems with non-Gaussian measurements, where a good initial guess can not be easily obtained for gradient flow solvers and the tightness of SDR does not hold for convex solvers.…”