广义相位可恢复矩阵框架的Lipschitz分析
Lipschitz Analysis of Generalized Phase Retrievable Matrix Frames
摘要：
经典的相位恢复问题出现在从语音识别到X射线晶体学和量子态断层扫描的范围内。矩阵框架的推广是自然的，因为它对应于不纯态的量子层析成像。我们提供了可计算的全局稳定性界的准线性分析映射$β$和理解相关问题的关键空间的微分几何的前进的道路。特别是，我们表现出惠特尼分层的低秩的半正定矩阵，使我们能够“分层”的计算的全球稳定性的界限。我们表明，对于不纯状态的情况下，没有这样的全局稳定性界可以获得非线性分析映射$α$相对于某些自然距离度量。最后，我们的计算全局较低的Lipschitz常数的$β$分析地图提供了新的条件，一个框架是广义相位可检索。
英文摘要：
The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to matrix frames is natural in the sense that it corresponds to quantum tomography of impure states. We provide computable global stability bounds for the quasi-linear analysis map $β$ and a path forward for understanding related problems in terms of the differential geometry of key spaces. In particular, we manifest a Whitney stratification of the positive semidefinite matrices of low rank which allows us to ``stratify'' the computation of the global stability bound. We show that for the impure state case no such global stability bounds can be obtained for the non-linear analysis map $α$ with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the $β$ analysis map provides novel conditions for a frame to be generalized phase retrievable.
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