  The phase retrieval problem is a fundamental problem in many fields, which is
appealing for investigation. It is to recover the signal vector
$\tilde{x}\in\mathbb{C}^d$ from a set of $N$ measurements
$b_n=|f^*_n\tilde{x}|^2,\ n=1,\cdots, N$, where $\{f_n\}_{n=1}^N$ forms a frame
of $\mathbb{C}^d$. %It is generally a non-convex minimization problem, which is
NP-hard. Existing algorithms usually use a least squares fitting to the
measurements, yielding a quartic polynomial minimization. In this paper, we
employ a new strategy by splitting the variables, and we solve a bi-variate
optimization problem that is quadratic in each of the variables. An alternating
gradient descent algorithm is proposed, and its convergence for any
initialization is provided. Since a larger step size is allowed due to the
smaller Hessian, the alternating gradient descent algorithm converges faster
than the gradient descent algorithm (known as the Wirtinger flow algorithm)
applied to the quartic objective without splitting the variables. Numerical
results illustrate that our proposed algorithm needs less iterations than
Wirtinger flow to achieve the same accuracy.
