  The problem of recovering a signal $\mathbf{x}\in \mathbb{R}^n$ from a set of
magnitude measurements $y_i=|\langle \mathbf{a}_i, \mathbf{x} \rangle |, \;
i=1,\ldots,m$ is referred as phase retrieval, which has many applications in
fields of physical sciences and engineering. In this paper we show that the
smoothed amplitude flow model for phase retrieval has benign geometric
structure under the optimal sampling complexity. In particular, we show that
when the measurements $\mathbf{a}_i\in \mathbb{R}^n$ are Gaussian random
vectors and the number of measurements $m\ge Cn$, our smoothed amplitude flow
model has no spurious local minimizers with high probability, ie., the target
solution $\mathbf{x}$ is the unique global minimizer (up to a global phase) and
the loss function has a negative directional curvature around each saddle
point. Due to this benign geometric landscape, the phase retrieval problem can
be solved by the gradient descent algorithms without spectral initialization.
Numerical experiments show that the gradient descent algorithm with random
initialization performs well even comparing with state-of-the-art algorithms
with spectral initialization in empirical success rate and convergence speed.
