  We study the phase retrieval problem, which solves quadratic system of
equations, i.e., recovers a vector $\boldsymbol{x}\in \mathbb{R}^n$ from its
magnitude measurements $y_i=|\langle \boldsymbol{a}_i, \boldsymbol{x}\rangle|,
i=1,..., m$. We develop a gradient-like 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
\cite{candes2015phase}, 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 $\boldsymbol{x}$. This improves the
sample complexity of WF, and achieves the same sample complexity as truncated
Wirtinger flow (TWF) \cite{chen2015solving}, 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.
