用Tanh Wirtinger流解随机二次方程组
Solving Random Systems of Quadratic Equations with Tanh Wirtinger Flow
摘要：
求解形式为$y_i = |a^T_i x|^2$、$i = 1, ..., m$和$x \in R^n$的n个变量和m个测量的二次方程组，也称为相位恢复，是一个困难的非凸问题。在标准高斯测量向量的情况下，Chen和Candes（2015）的双曲正切流算法是一种有效的解决方案。本文提出了一种新的谱初始化算法，并给出了一种新的谱初始化算法。我们证明了新的离散流和初始化方法实现了线性采样和计算复杂性。我们进一步扩展了新的定相算法，将其与其他现有的方法相结合。最后，我们证明了我们的新方法的有效性，在低的数据参数比设置的测量数小于信息理论的限制，即$m < 2n$，通过数值试验。例如，我们的方法可以解决具有高斯测量向量的二次方程组，当$m/n = 1.7$和$n = 1000$时概率为$\ge 97\%$，当$m/n = 1.5$和$n = 1000$时概率为$\approx 60\%$。
英文摘要：
Solving quadratic systems of equations in n variables and m measurements of the form $y_i = |a^T_i x|^2$ , $i = 1, ..., m$ and $x \in R^n$ , which is also known as phase retrieval, is a hard nonconvex problem. In the case of standard Gaussian measurement vectors, the wirtinger flow algorithm Chen and Candes (2015) is an efficient solution. In this paper, we proposed a new form of wirtinger flow and a new spectral initialization method based on this new algorithm. We proved that the new wirtinger flow and initialization method achieve linear sample and computational complexities. We further extended the new phasing algorithm by combining it with other existing methods. Finally, we demonstrated the effectiveness of our new method in the low data to parameter ratio settings where the number of measurements which is less than information-theoretic limit, namely, $m < 2n$, via numerical tests. For instance, our method can solve the quadratic systems of equations with gaussian measurement vector with probability $\ge 97\%$ when $m/n = 1.7$ and $n = 1000$, and with probability $\approx 60\%$ when $m/n = 1.5$ and $n = 1000$.
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。
