We tackle the problem of recovering a complex signal x∈Cn{\boldsymbol{x}}\in \mathbb{C}^n from quadratic measurements of the form yi=x∗Aixy_i={\boldsymbol{x}}^*{\boldsymbol{A}}_i{\boldsymbol{x}}, where Ai{\boldsymbol{A}}_i is a full-rank, complex random measurement matrix whose entries are generated from a rotation-invariant sub-Gaussian distribution. We formulate it as the minimization of a nonconvex loss. This problem is related to the well understood phase retrieval problem where the measurement matrix is a rank-1 positive semidefinite matrix. Here we study the general full-rank case which models a number of key applications such as molecular geometry recovery from distance distributions and compound measurements in phaseless diffractive imaging. Most prior works either address the rank-1 case or focus on real measurements. The several papers that address the full-rank complex case adopt the computationally-demanding semidefinite relaxation approach. In this paper we prove that the general class of problems with rotation-invariant sub-Gaussian measurement models can be efficiently solved with high probability via the standard framework comprising a spectral initialization followed by iterative Wirtinger flow updates on a nonconvex loss. Numerical experiments on simulated data corroborate our theoretical analysis.

中文翻译：

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用全秩随机矩阵求解复杂二次系统


我们解决从 yi=x*Aixy_i={\boldsymbol{x}}^*{ 形式的二次测量中恢复复数信号 x∈Cn{\boldsymbol{x}}\in \mathbb{C}^n 的问题\boldsymbol{A}}_i{\boldsymbol{x}}，其中 Ai{\boldsymbol{A}}_i 是满秩、复数随机测量矩阵，其条目是从旋转不变的亚高斯分布生成的。我们将其表述为非凸损失的最小化。该问题与众所周知的相位检索问题相关，其中测量矩阵是 1 阶正半定矩阵。在这里，我们研究一般的满秩情况，它模拟了许多关键应用，例如从距离分布恢复分子几何形状以及无相衍射成像中的化合物测量。大多数先前的工作要么解决排名 1 的情况，要么专注于实际测量。解决满秩复杂情况的几篇论文采用了计算要求较高的半定松弛方法。在本文中，我们证明旋转不变亚高斯测量模型的一般问题可以通过标准框架以高概率有效解决，该标准框架包括谱初始化，然后对非凸损失进行迭代 Wirtinger 流更新。模拟数据的数值实验证实了我们的理论分析。