  The phase retrieval problem is concerned with recovering an unknown signal
$\bf{x} \in \mathbb{R}^n$ from a set of magnitude-only measurements
$y_j=|\langle \bf{a}_j,\bf{x} \rangle|, \; j=1,\ldots,m$. A natural least
squares formulation can be used to solve this problem efficiently even with
random initialization, despite its non-convexity of the loss function. One way
to explain this surprising phenomenon is the benign geometric landscape: (1)
all local minimizers are global; and (2) the objective function has a negative
curvature around each saddle point and local maximizer. In this paper, we show
that $m=O(n \log n)$ Gaussian random measurements are sufficient to guarantee
the loss function of a commonly used estimator has such benign geometric
landscape with high probability. This is a step toward answering the open
problem given by Sun-Qu-Wright, in which the authors suggest that $O(n \log n)$
or even $O(n)$ is enough to guarantee the favorable geometric property.
