A fundamental task in phase retrieval is to recover an unknown signal $\vx\in \Rn$ from a set of magnitude-only measurements $y_i=\abs{\nj{\va_i,\vx}}, \; i=1,\ldots,m$. In this paper, we propose two novel perturbed amplitude models (PAMs) which have non-convex and quadratic-type loss function. When the measurements $ \va_i \in \Rn$ are Gaussian random vectors and the number of measurements $m\ge Cn$, we rigorously prove that the PAMs admit no spurious local minimizers with high probability, i.e., the target solution $ \vx$ is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Thanks to the well-tamed benign geometric landscape, one can employ the vanilla gradient descent method to locate the global minimizer $\vx$ (up to a global phase) without spectral initialization. We carry out extensive numerical experiments to show that the gradient descent algorithm with random initialization outperforms state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.

翻译：阶段检索的一项基本任务是从一个量级测量器中恢复一个未知的信号$vxxxxx(单位:$y_i ⁇ abs=nj_va_i,\vx ⁇,\;i=1,\ldots,m$。在本文中,我们提出两个新的扰动振动模型(PAMs),这些模型具有非电解仪和四面形损失功能。当测量仪$\va_i\in\rn$是高斯随机矢量和测量数量$m\geCn$时,我们严格地证明,PAMs接受的本地最小度测量器没有高概率,也就是说,目标解决方案$\x$是独特的全球最小度模型(全球阶段之前),而损失函数在每一个马鞍周围有一个负向向曲线曲线曲线曲线曲线曲线曲线。由于精心设计的精确的测深度,所以可以使用香草梯梯梯梯梯梯梯梯梯梯梯梯梯梯梯梯底系定位全球最小值 $\xx$(直到全球水平),我们严格地证明,Paslxlxlxlxlation 的初始级趋同级模型化模型化实验性演化的初始演算法,我们将在初始进度上进行广泛的演算。