  In this paper, we provide a fine-grained analysis of the local landscape of
phase retrieval under the regime with limited samples. Our aim is to ascertain
the minimal sample size necessary to guarantee a benign local landscape
surrounding global minima in high dimensions. Let $n$ and $d$ denote the sample
size and input dimension, respectively. We first explore the local convexity
and establish that when $n=o(d\log d)$, for almost every fixed point in the
local ball, the Hessian matrix must have negative eigenvalues as long as $d$ is
sufficiently large. Consequently, the local landscape is highly non-convex. We
next consider the one-point strong convexity and show that as long as
$n=\omega(d)$, with high probability, the landscape is one-point strongly
convex in the local annulus: $\{w\in\mathbb{R}^d: o_d(1)\leqslant
\|w-w^*\|\leqslant c\}$, where $w^*$ is the ground truth and $c$ is an absolute
constant. This implies that gradient descent initialized from any point in this
domain can converge to an $o_d(1)$-loss solution exponentially fast.
Furthermore, we show that when $n=o(d\log d)$, there is a radius of
$\widetilde\Theta\left(\sqrt{1/d}\right)$ such that one-point convexity breaks
in the corresponding smaller local ball. This indicates an impossibility to
establish a convergence to exact $w^*$ for gradient descent under limited
samples by relying solely on one-point convexity.
