Optimal Spectral Initialization for Signal Recovery With Applications to Phase Retrieval
Optimal Spectral Initialization for Signal Recovery With Applications to Phase Retrieval
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DOI：
10.1109/tsp.2019.2904918
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发表时间：
2019-05-01
影响因子：
5.4
通讯作者：
Lu, Yue M.
中科院分区：
文献类型：
--
作者：
Luo, Wangyu;Alghamdi, Wael;Lu, Yue M.
我们提出了广泛用于初始化非凸优化算法的谱方法的优化设计，以解决相位恢复和其他信号恢复问题。本文利用最新的结果，提供了高维限制下谱方法性能的精确表征。这种表征使我们能够将优化设计任务映射到加权 L-2 函数空间中的约束优化问题。后者有一个封闭式的解决方案。有趣的是，在温和的技术条件下，我们的结果表明存在一种固定设计，该设计在所有采样率上都是一致最优的。数值模拟证明了所提出的优化设计相对于文献中现有结构所带来的性能改进。在最近的一项工作中，蒙德里和蒙塔纳里表明存在一个弱恢复阈值，低于该阈值光谱方法无法提供有用的估计。我们的结果通过推导光谱方法的基本极限超出上述阈值来补充这项工作。
We present the optimal design of a spectral method widely used to initialize nonconvex optimization algorithms for solving phase retrieval and other signal recovery problems. This paper leverages recent results that provide an exact characterization of the performance of the spectral method in the high-dimensional limit. This characterization allows us to map the task of optimal design to a constrained optimization problem in a weighted L-2 function space. The latter has a closed-form solution. Interestingly, under a mild technical condition, our results show that there exists a fixed design that is uniformly optimal over all sampling ratios. Numerical simulations demonstrate the performance improvement brought by the proposed optimal design over existing constructions in the literature. In a recent work, Mondelli and Montanari have shown the existence of a weak recovery threshold below which the spectral method cannot provide useful estimates. Our results serve to complement that work by deriving the fundamental limit of the spectral method beyond the aforementioned threshold.