Reshaped Wirtinger Flow and Incremental Algorithm for Solving Quadratic System of Equations

Abstract

We study the phase retrieval problem, which solves quadratic system of equations, i.e., recovers a vector x ∈ R n from its magnitude measurements yi = | ai, x |, i = 1, ..., m. We develop a gradientlike algorithm (referred to as RWF representing reshaped Wirtinger flow) by minimizing a nonconvex nonsmooth loss function. In comparison with existing nonconvex Wirtinger flow (WF) algorithm [1], although the loss function becomes nonsmooth, it involves only the second power of variable and hence reduces the complexity. We show that for random Gaussian measurements, RWF enjoys geometric convergence to a global optimal point as long as the number m of measurements is on the order of n, the dimension of the unknown x. This improves the sample complexity of WF, and achieves the same sample complexity as truncated Wirtinger flow (TWF) [2], but without truncation in gradient loop. Furthermore, RWF costs less computationally than WF, and runs faster numerically than both WF and TWF. We further develop the incremental (stochastic) reshaped Wirtinger flow (IRWF) and show that IRWF converges linearly to the true signal. We further establish performance guarantee of an existing Kaczmarz method for the phase retrieval problem based on its connection to IRWF. We also empirically demonstrate that IRWF outperforms existing ITWF algorithm (stochastic version of TWF) as well as other batch algorithms.

Phase retrieval via Sparse Wirtinger Flow

“…For Poisson likelihood function, truncated Wirtinger flow(TWF) was proposed [14]. For amplitude based models, reshaped WF and TAF(Truncated Amplituded Flow) were considered in [15] [16]. Those methods often need m/n > 3 for the exactly recovery.…”

Low-Rank Phase Retrieval

“…WF also has a slower convergence rate. Two recent modifications of TWF [22], [23] have the same order complexities but improved empirical performance.…”

Robust Wirtinger Flow for Phase Retrieval with Arbitrary Corruption

“…In the literature of physical sciences, the problem of solving (1.1) is also known as phase retrieval (Fienup, 1978;Candes et al, 2015b), where the goal is to reconstruct the unknown signal vector from magnitude only measurements. There exists a large body of literature (Fienup, 1978(Fienup, , 1982Gerchberg, 1972;Candes et al, 2013Candes et al, , 2015aNetrapalli et al, 2013;Candes et al, 2015b;Goldfarb and Qin, 2014;Wei, 2015;Zhang and Liang, 2016;Wang et al, 2016b;Sun et al, 2016;Wang et al, 2016a;Zhang et al, 2016b;Huang et al, 2016;Goldstein and Studer, 2017) for phase retrieval in the noise-free and noisy cases. The applications of phase retrieval include X-ray crystallography (Harrison, 1993;Miao et al, 1999), microscopy (Miao et al, 2008), diffraction and array imaging (Bunk et al, 2007;Chai et al, 2010), optics (Millane, 1990) and so on.…”