A geometric analysis of phase retrieval

Abstract

Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, y k = |a * k x| for k = 1, . . . , m, is it possible to recover x ∈ C n (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines, and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretical explanations. In this paper, we take a step towards bridging this gap. We prove that when the measurement vectors a k 's are generic (i.i.d. complex Gaussian) and numerous enough (m ≥ Cn log 3 n), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) there are no spurious local minimizers, and all global minimizers are equal to the target signal x, up to a global phase; and (2) the objective function has a negative directional curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.Provable methods for GPR. Although heuristic methods for GPR have been used effectively in practice [GS72, Fie82, SEC + 15, JEH15], only recently have researchers begun to develop methods with provable performance guarantees. The first results of this nature were obtained using semidefinite programming (SDP) relaxations [CESV13, CSV13, CL14, CLS15a, WdM15, VX14]. While this represented a substantial advance in theory, the computational complexity of semidefinite programming limits the practicality of this approach. 7 Recently, several provable nonconvex methods have been proposed for GPR.[NJS13] augmented the seminal error-reduction method [GS72] with spectral initialization and resampling to obtain the first provable nonconvex method for GPR. [CLS15b] studied the nonconvex formulation (1.1) under the same hypotheses as this paper, and showed that a combination of spectral initialization and local gradient descent recovers the true signal with near-optimal sample complexity. [CC15] worked with a different nonconvex formulation, and refined the spectral initialization and the local gradient descent with a step-adaptive truncation. With the modifications, they reduced the sample requirement to the optimal order. 8 More recent work in this line [ZCL16, ZL16, WGE16, KÖ16, GX16, BE16, Wal16] concerns error stability, alternative formulations, algorithms, and measurement models. Compared to the SDP-based methods, these methods are more scalable and closer to methods used in practice. All these analyses are based on local geometry in nature, and hence depend on the spectral initializer being sufficiently close to the target set. In contrast, we explicitly characterize the global function landscape of (1.1). Its benign global geometric structure allows several algorithmic choices (see Section 1.3) that need no special init...

Convolutional Phase Retrieval via Gradient Descent

“…al. [36] reveal that the nonconvex objective (1.5) actually has a benign global geometry: with high probability, it has no bad critical points with m ≥ Ω(n log 3 n) samples 4 . Such a result enables initialization-free nonconvex recovery 5 [42,43].…”

The Projected Power Method: An Efficient Algorithm for Joint Alignment from Pairwise Differences

“…Despite the general intractability, recent years have seen progress on nonconvex procedures for several classes of problems, including low-rank matrix recovery [24,45,50,51,56,64,75,80,84,89,90], phase retrieval [11,13,17,25,56,61,68,74,81,86,87], dictionary learning [72,73], blind deconvolution [54,56], and empirical risk minimization [58], to name just a few. For example, we have learned that several problems of this kind provably enjoy benign geometric structure when the sample complexity is sufficiently large in the sense that all local stationary points (except for the global optimum) become saddle points and are not difficult to escape [7,34,53,72,73].…”

“…For example, we have learned that several problems of this kind provably enjoy benign geometric structure when the sample complexity is sufficiently large in the sense that all local stationary points (except for the global optimum) become saddle points and are not difficult to escape [7,34,53,72,73]. For the problem of solving certain random systems of quadratic equations, this phenomenon arises as long as the number of equations or sample size exceeds the order of n log 3 n, with n denoting the number of unknowns [74]. 1 We have also learned that it is possible to minimize certain nonconvex random functionals-closely associated with the famous phase retrieval problem-even when there may be multiple local minima [13,17].…”

VIPR: Vectorial Implementation of Phase Retrieval for fast and accurate microscopic pixel-wise pupil estimation

“…Phase retrieval can be viewed as a nonconvex optimization problem [37]. In general, such optimization problems pose a challenge as they can have spurious local minimizers, saddle points, and significant sensitivity to good initialization; However, it has been shown that PR can achieve convergence using Stochastic Gradient Descent (SGD) [18].…”