Phase Retrieval via Reweighted Amplitude Flow
Phase Retrieval via Reweighted Amplitude Flow
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DOI：
10.1109/tsp.2018.2818077
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发表时间：
2018-06
影响因子：
5.4
通讯作者：
G. Wang;G. Giannakis;Y. Saad;Jie Chen
中科院分区：
文献类型：
--
作者：
G. Wang;G. Giannakis;Y. Saad;Jie Chen
本文讨论了一个二次方程组y_i= x的n维解|\langle \boldsymbol {a}_i，\boldsymbol {x}\rangle| ^2 $ for $1\leq i \leq m$，这也被称为广义相位恢复问题。对于这个NP难问题，一种新的方法是开发用于最小化基于幅度的最小二乘经验损失，它开始与加权最大相关初始化可通过几个电源或Lanczos迭代，随后通过连续的改进的基础上的一系列迭代重新加权梯度迭代。这两个阶段（初始化和梯度流）通过包含一个新的（重新）加权正则化过程将自己与先前的贡献区分开来。对于某些随机测量模型，新的计划被证明能够恢复的真实解决方案$\boldsymbol {x}$的时间成正比的阅读数据$\lbrace（\boldsymbol {a}_i;y_i）\rbrace _{1\leq i \leq m}$。这具有很高的概率，并且不需要对要恢复的信号矢量$\boldsymbol {x}$进行额外的假设，只要方程的数量$m$是信号矢量中未知数的数量$n$的某个常数$c>0$倍，即$m>cn$。从经验上讲，这种贡献的结果是：第一，（几乎）$\text{100}{\%}$完美的信号恢复在高维（说$n\geq 2000$）政权只给出了一个信息理论的限制数量的无噪声方程，即$m=2n-1$，在真实的高斯的情况下;和第二，（几乎）最佳的统计精度在存在有界支持的加性噪声。最后，大量的数值试验，使用合成数据和真实的图像证实显着提高恢复性能和计算效率的新方案相对于国家的最先进的方法。
This paper deals with finding an $n$ -dimensional solution $\boldsymbol {x}$ to a system of quadratic equations of the form $y_i=|\langle \boldsymbol {a}_i,\boldsymbol {x}\rangle |^2$ for $1\leq i \leq m$, which is also known as the generalized phase retrieval problem. For this NP-hard problem, a novel approach is developed for minimizing the amplitude-based least-squares empirical loss, which starts with a weighted maximal correlation initialization obtainable through a few power or Lanczos iterations, followed by successive refinements based on a sequence of iteratively reweighted gradient iterations. The two stages (initialization and gradient flow) distinguish themselves from prior contributions by the inclusion of a fresh (re)weighting regularization procedure. For certain random measurement models, the novel scheme is shown to be able to recover the true solution $\boldsymbol {x}$ in time proportional to reading the data $\lbrace (\boldsymbol {a}_i;y_i)\rbrace _{1\leq i \leq m}$. This holds with high probability and without extra assumption on the signal vector $\boldsymbol {x}$ to be recovered, provided that the number $m$ of equations is some constant $c>0$ times the number $n$ of unknowns in the signal vector, namely $m>cn$ . Empirically, the upshots of this contribution are: first, (almost) $\text{100}{\%}$ perfect signal recovery in the high-dimensional (say $n\geq 2000$) regime given only an information-theoretic limit number of noiseless equations, namely $m=2n-1$, in the real Gaussian case; and second, (nearly) optimal statistical accuracy in the presence of additive noise of bounded support. Finally, substantial numerical tests using both synthetic data and real images corroborate markedly improved recovery performance and computational efficiency of the novel scheme relative to the state-of-the-art approaches.