We consider the problem of recovering a real-valued $n$-dimensional signal from $m$ phaseless, linear measurements and analyze the amplitude-based non-smooth least squares objective. We establish local convergence of gradient descent with optimal sample complexity based on the uniform concentration of a random, discontinuous matrix-valued operator arising from the objective's gradient dynamics. While common techniques to establish uniform concentration of random functions exploit Lipschitz continuity, we prove that the discontinuous matrix-valued operator satisfies a uniform matrix concentration inequality when the measurement vectors are Gaussian as soon as $m = \Omega(n)$ with high probability. We then show that satisfaction of this inequality is sufficient for gradient descent with proper initialization to converge linearly to the true solution up to the global sign ambiguity. As a consequence, this guarantees local convergence for Gaussian measurements at optimal sample complexity. The concentration methods in the present work have previously been used to establish recovery guarantees for a variety of inverse problems under generative neural network priors. This paper demonstrates the applicability of these techniques to more traditional inverse problems and serves as a pedagogical introduction to those results.

本文考虑从m个线性测量中恢复n维实值信号的问题，建立了基于梯度动力学随机不连续矩阵的集中方法，分析了基于振幅的非平滑最小二乘目标函数的局部收敛性和最优样本复杂度，证明了当观测向量为高斯测量时，不连续矩阵满足均匀矩阵集中不等式的可能性非常高，从而足以保证 负梯度下降法以正确的初始化线性收敛到真实解，文中结果对于生成类神经网络以外的问题具有普遍适用性

非Lipschitz矩阵集中下的幅度流次梯度下降算法的最优采样复杂度