Reshaped Wirtinger Flow for Solving Quadratic System of Equations

We study the problem of recovering a vector $\bx\in \bbR^n$ from its magnitude measurements $y_i=|\langle \ba_i, \bx\rangle|, i=1,..., m$. Our work is along the line of the Wirtinger flow (WF) approach \citet{candes2015phase}, which solves the problem by minimizing a nonconvex loss function via a gradient algorithm and can be shown to converge to a global optimal point under good initialization. In contrast to the smooth loss function used in WF, we adopt a nonsmooth but lower-order loss function, and design a gradient-like algorithm (referred to as reshaped-WF). We show that for random Gaussian measurements, reshaped-WF enjoys geometric convergence to a global optimal point as long as the number $m$ of measurements is at the order of $\cO(n)$, where $n$ is the dimension of the unknown $\bx$. This improves the sample complexity of WF, and achieves the same sample complexity as truncated-WF \citet{chen2015solving} but without truncation at gradient step. Furthermore, reshaped-WF costs less computationally than WF, and runs faster numerically than both WF and truncated-WF. Bypassing higher-order variables in the loss function and truncations in the gradient loop, analysis of reshaped-WF is simplified.