  Can we recover a complex signal from its Fourier magnitudes? More generally,
given a set of $m$ measurements, $y_k = |\mathbf a_k^* \mathbf x|$ for $k = 1,
\dots, m$, is it possible to recover $\mathbf x \in \mathbb{C}^n$ (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 $\mathbf a_k$'s are generic (i.i.d. complex Gaussian) and the number of
measurements is large enough ($m \ge C n \log^3 n$), 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 $\mathbf x$, up to a global phase;
and (2) the objective function has a negative 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.
