This paper explores the problem of generalized phase retrieval, which involves reconstructing a length-$n$ signal $\bm{x}$ from its $m$ phaseless samples $y_k = \left|\langle \bm{a}_k,\bm{x}\rangle\right|^2$, where $k = 1,2,...,m$, and $\bm{a}_k$ are the measurement vectors. This problem can be reformulated into recovering a positive semidefinite rank-$1$ matrix $\bm{X}=\bm{x}\bm{x}^*$ from linear samples $\bm{y}=\mathcal{A}(\bm{X})\in\mathbb{R}^m$, thereby requiring us to find a rank-$1$ solution of the linear equations. We demonstrate that several existing phase retrieval algorithms, including Wirtinger Flow (WF) and the canonical Riemannian gradient descent (RGD), actually solve the least-squares fitting of this linear equation on the Riemannian manifold of rank-$1$ matrices, but utilize different metrics on this manifold. Nevertheless, these metrics only allow for a stable and far-apart-from-isometric embedding of rank-$1$ matrices to $\mathbb{R}^m$ by $\mathcal{A}$, resulting in a linear convergence with a considerably large convergence factor. To expedite the convergence, we establish a new metric on the rank-$1$ matrix manifold that facilitates the nearly isometric embedding of rank-$1$ matrices into $\mathbb{R}^m$ through $\mathcal{A}$. A RGD algorithm under this new metric, termed Weighted RGD (WRGD), is proposed to tackle the phase retrieval problem. Owing to the near isometry, we prove that our WRGD algorithm, initialized by spectral methods, can linearly converge to the underlying signal $\bm{x}$ with a small convergence factor. Empirical experiments strongly validate the efficiency and resilience of our algorithms compared to the truncated Wirtinger Flow (TWF) algorithm and the canonical RGD algorithm.

翻译：本文探讨广义相位恢复问题，该问题涉及从$m$个无相位样本$y_k = \left|\langle \bm{a}_k,\bm{x}\rangle\right|^2$（其中$k = 1,2,...,m$，$\bm{a}_k$为测量向量）中重构长度为$n$的信号$\bm{x}$。该问题可转化为从线性样本$\bm{y}=\mathcal{A}(\bm{X})\in\mathbb{R}^m$中恢复半正定秩$1$矩阵$\bm{X}=\bm{x}\bm{x}^*$，从而要求我们在线性方程组中寻找秩$1$解。我们证明，包括Wirtinger Flow (WF)和经典黎曼梯度下降(RGD)在内的几种现有相位恢复算法，实际上是在秩$1$矩阵的黎曼流形上求解该线性方程的最小二乘拟合，但采用了该流形上的不同度量。然而，这些度量仅能通过$\mathcal{A}$实现秩$1$矩阵到$\mathbb{R}^m$的稳定但远离等距的嵌入，导致线性收敛但收敛因子相当大。为加速收敛，我们在秩$1$矩阵流形上建立了一种新度量，该度量有助于通过$\mathcal{A}$将秩$1$矩阵近乎等距嵌入到$\mathbb{R}^m$中。基于这种新度量，我们提出了一种RGD算法，称为加权RGD (WRGD)，用于解决相位恢复问题。由于具有近似等距性，我们证明通过谱方法初始化的WRGD算法能够以较小的收敛因子线性收敛到原始信号$\bm{x}$。实验证据充分验证了我们的算法相较于截断Wirtinger Flow (TWF)算法和经典RGD算法的效率和鲁棒性。