Solving Complex Quadratic Equations with Full-rank Random Gaussian Matrices
Solving Complex Quadratic Equations with Full-rank Random Gaussian Matrices
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DOI：
10.1109/icassp.2019.8683280
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发表时间：
2019-05
期刊：
影响因子：
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通讯作者：
Shuai Huang;Sidharth Gupta;Ivan Dokmanić
中科院分区：
文献类型：
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作者：
Shuai Huang;Sidharth Gupta;Ivan Dokmanić
我们解决了从形式y = x ∗ aix的二次测量中恢复复杂信号的问题，其中$ \ left \ {{{{{{\ mathbf {a}} _ i}}}}}}} 1}^m $是一组复杂的IID标准高斯矩阵。模型许多关键应用程序例如，从距离分布中恢复的分子几何形状和无量衍射成像中的化合物测量值。因此，在本文中，计算要求。我们证明，当测量数量超过信号的长度时，可以从复杂的二次测量值中恢复全球最佳解决方案，并具有很高的概率。
We tackle the problem of recovering a complex signal x ∈ ℂn from quadratic measurements of the form y = x∗Aix, where $\left\{ {{{\mathbf{A}}_i}} \right\}_{i = 1}^m$ is a set of complex iid standard Gaussian matrices. This non-convex problem is related to the well understood phase retrieval problem where Ai is a rank-1 positive semidefinite matrix. Here we study a 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 work either addresses the rank-1 case or focuses on real measurements. The several papers that address the full-rank complex case adopt the semidefinite relaxation approach and are thus computationally demanding. In this paper we propose a method based on the standard framework comprising a spectral initialization followed by iterative gradient descent updates. We prove that when the number of measurements exceeds the signal’s length by some constant factor, a globally optimal solution can be recovered from complex quadratic measurements with high probability. Numerical experiments on simulated data corroborate our theoretical analysis.