The Local Landscape of Phase Retrieval Under Limited Samples

In this paper, we present a fine-grained analysis of the local landscape of phase retrieval under the regime of limited samples. Specifically, we aim to ascertain the minimal sample size required 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 has negative eigenvalues, provided $d$ is sufficiently large. % Consequently, the local landscape is highly non-convex. We next consider the one-point 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 down in the corresponding smaller local ball. This indicates an impossibility of establishing a convergence to the exact $w^*$ for gradient descent under limited samples by relying solely on one-point convexity.