The Local Landscape of Phase Retrieval Under Limited Samples

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. 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 to establish a convergence to the exact $w^{*}$ for gradient descent under limited samples by relying solely on one-point convexity.