Phase Retrieval Using Alternating Minimization

Abstract

Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. More than four decades after it was first proposed, the seminal error reduction algorithm of Gerchberg and Saxton [21] and Fienup [19] is still the popular choice for solving many variants of this problem. The algorithm is based on alternating minimization; i.e. it alternates between estimating the missing phase information, and the candidate solution. Despite its wide usage in practice, no global convergence guarantees for this algorithm are known. In this paper, we show that a (resampling) variant of this approach converges geometrically to the solution of one such problem -finding a vector x from y, A, where y = |A T x| and |z| denotes a vector of element-wise magnitudes of z -under the assumption that A is Gaussian.Empirically, we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem (which are based on "lifting" to a convex matrix problem) in sample complexity and robustness to noise. However, it is much more efficient and can scale to large problems. Analytically, for a resampling version of alternating minimization, we show geometric convergence to the solution, and sample complexity that is off by log factors from obvious lower bounds. We also establish close to optimal scaling for the case when the unknown vector is sparse. Our work represents the first theoretical guarantee for alternating minimization (albeit with resampling) for any variant of phase retrieval problems in the non-convex setting.

Community recovery in hypergraphs

“…Detailed procedures are presented in Algorithm 1. Our algorithm is inspired by two-stage approaches that have been applied to a wide variety of problems including matrix completion [57], [58], phase retrieval [59], [60], robust PCA [61], community recovery [18], [20], [32], [62], [63], EM-algorithm [64], and rank aggregation [65]. …”

PhaseCode: Fast and Efficient Compressive Phase Retrieval Based on Sparse-Graph Codes

“…While algorithms based on SDP provide theoretical performance guarantees and are robust to noise, they suffer from a high computational complexity of O(n 3 ) rendering them unsuited for many practical applications that require n to scale. 3 In [16], the authors propose an algorithm based on alternating minimization that reconstructs the signal with Θ(n log(n)…”

“…The PhaseLift method is also proposed for the sparse case in [14] and [17], requiring Θ(K 2 log(n)) intensity measurements, and having a computational complexity of O(n 3 ), making the method less practical for large-scale applications. In [33], the authors propose an efficient algorithm based on polarization method that is able to stably reconstruct any K-sparse vector from Θ(K log(n)) noisy intensity measurements with complexity polynomial in n. The alternating minimization method in [16] can also be adapted to the sparse case with Θ(K 2 log(n)) measurements and a complexity of O(K 3 n log(n)). Compressive phase retrieval via generalized approximate message passing (PR-GAMP) is proposed in [13], with good performance in both runtime and noise robustness shown via simulations without theoretical justification.…”

Comparison of non-convex cost functionals for the consideration of phase differences in phaseless near-field far-field transformations of measured antenna fields

“…The initial algorithm has been further developed throughout the years (for example in Fienup, 1982) and the topic is still under heavy research. In the last decade, one has often tried to retrieve the phase via non-convex minimization (Netrapalli et al, 2013;Candes et al, 2015;Zhang and Liang, 2016) or by a convex relaxation of the non-convex formulation (Waldspurger et al, 2015;Yurtsever et al, 2015;Candes et al, 2013;Bauschke et al, 2002). Because of their high numerical complexity, convex relaxations are suitable for small and medium sized problems only.…”

“…Non-convex minimizations suffer from stationary points of the cost functional. Convergence guarantees have been established for the non-convex minimization in Candes et al (2015); Zhang and Liang (2016) and Netrapalli et al (2013). However, the convergence criteria hold only, if enough measurements can be obtained, which follow certain probability distributions.…”