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In this paper, we provide a fine-grained analysis of the local landscape ofphase retrieval under the regime with limited samples. Our aim is to ascertainthe minimal sample size necessary to guarantee a benign local landscapesurrounding global minima in high dimensions. Let $n$ and $d$ denote the samplesize and input dimension, respectively. We first explore the local convexityand establish that when $n=o(d\log d)$, for almost every fixed point in thelocal ball, the Hessian matrix must have negative eigenvalues as long as $d$ issufficiently large. Consequently, the local landscape is highly non-convex. Wenext consider the one-point strong convexity and show that as long as$n=\omega(d)$, with high probability, the landscape is one-point stronglyconvex 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 absoluteconstant. This implies that gradient descent initialized from any point in thisdomain 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 breaksin the corresponding smaller local ball. This indicates an impossibility toestablish a convergence to exact $w^*$ for gradient descent under limitedsamples by relying solely on one-point convexity.

The Local Landscape of Phase Retrieval Under Limited Samples