用随机初始猜测法进行相位恢复的效果与谱初始化法相当
Solving phase retrieval with random initial guess is nearly as good as by spectral initialization
摘要：
从一组幅度测量值$y_i=|\langle \mathbf{a}_i, \mathbf{x} \rangle |, \; i=1,\ldots,m$恢复信号$\mathbf{x}\in \mathbb{R}^n$的问题被称为相位恢复，其在物理科学和工程领域中具有许多应用。本文证明了在最佳采样复杂度下，用于相位恢复的平滑振幅流模型具有良好的几何结构。特别地，我们证明了当测量值$\mathbf{a}_i\in \mathbb{R}^n$是高斯随机向量并且测量值的数目为$m\ge Cn$时，我们的平滑振幅流模型没有具有高概率的伪局部极小值，即，目标解$\mathbf{x}$是唯一的全局最小值（直到全局阶段），并且损失函数在每个鞍点周围具有负方向曲率。由于这种良好的几何景观，相位恢复问题可以通过梯度下降算法来解决，而无需频谱初始化。数值实验表明，随机初始化的梯度下降算法在经验成功率和收敛速度上与谱初始化算法相比，具有更好的性能.
英文摘要：
The problem of recovering a signal $\mathbf{x}\in \mathbb{R}^n$ from a set of magnitude measurements $y_i=|\langle \mathbf{a}_i, \mathbf{x} \rangle |, \; i=1,\ldots,m$ is referred as phase retrieval, which has many applications in fields of physical sciences and engineering. In this paper we show that the smoothed amplitude flow model for phase retrieval has benign geometric structure under the optimal sampling complexity. In particular, we show that when the measurements $\mathbf{a}_i\in \mathbb{R}^n$ are Gaussian random vectors and the number of measurements $m\ge Cn$, our smoothed amplitude flow model has no spurious local minimizers with high probability, ie., the target solution $\mathbf{x}$ is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Due to this benign geometric landscape, the phase retrieval problem can be solved by the gradient descent algorithms without spectral initialization. Numerical experiments show that the gradient descent algorithm with random initialization performs well even comparing with state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
