论文标题
确切的非凸相检索的确定性理论
A Deterministic Theory for Exact Non-Convex Phase Retrieval
论文作者
论文摘要
在本文中，我们分析了连接器流量（WF）的非凸框架以进行相检索，并通过低级矩阵恢复理论的镜头确定了一种新颖的条件，以实现通用精确恢复。通过抬高域中的透视图，我们表明，在升起的向前模型的单个条件下，WF迭代到真实解决方案的收敛性。结果，得出了光谱初始化的精度与{正常性条件}的有效性之间的确定性关系。特别是，我们确定光谱矩阵上的某些浓度特性必须均匀地保持，并具有足够紧密的常数。这最终达到了足够的条件，等于等级1的限制等轴测型属性，正向半明确矩阵，而在提升的正向模型上的需求较小，而不是在文献中出色的低阶 - 矩阵重新发现方法的要求。我们通过新颖的界限和信噪比的限制来表征我们框架的性能限制，以使理论保证在适当的样品复杂性下使用频谱初始化有效。
In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we show that the convergence of the WF iterates to a true solution is attained geometrically under a single condition on the lifted forward model. As a result, a deterministic relationship between the accuracy of spectral initialization and the validity of {the regularity condition} is derived. In particular, we determine that a certain concentration property on the spectral matrix must hold uniformly with a sufficiently tight constant. This culminates into a sufficient condition that is equivalent to a restricted isometry-type property over rank-1, positive semi-definite matrices, and amounts to a less stringent requirement on the lifted forward model than those of prominent low-rank-matrix-recovery methods in the literature. We characterize the performance limits of our framework in terms of the tightness of the concentration property via novel bounds on the convergence rate and on the signal-to-noise ratio such that the theoretical guarantees are valid using the spectral initialization at the proper sample complexity.
