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.

翻译：本文对有限样本条件下相位恢复问题的局部景观进行了精细分析。具体而言，我们旨在确定保证高维全局极小值点附近存在良性局部景观所需的最小样本量。令 $n$ 和 $d$ 分别表示样本量和输入维度。我们首先探究局部凸性，并证明当 $n=o(d\log d)$ 时，对于局部球内几乎任意固定点，若 $d$ 充分大，其Hessian矩阵均存在负特征值。% 因此，局部景观具有高度非凸性。接着我们考虑单点凸性，证明只要 $n=\omega(d)$，以高概率在局部环形区域 $\{w\in\mathbb{R}^d: o_d(1)\leqslant \|w-w^*\|\leqslant c\}$ 内景观是单点强凸的，其中 $w^*$ 为真实参数，$c$ 为绝对常数。这表明从该区域内任意点初始化的梯度下降法都能以指数速度收敛到 $o_d(1)$ 损失解。此外，我们证明当 $n=o(d\log d)$ 时，存在半径为 $\widetilde\Theta\left(\sqrt{1/d}\right)$ 的邻域使得相应更小局部球内的单点凸性被破坏。这意味着在有限样本条件下，仅依靠单点凸性无法保证梯度下降收敛到精确解 $w^*$。