Lipschitz Analysis of Generalized Phase Retrievable Matrix Frames
Lipschitz Analysis of Generalized Phase Retrievable Matrix Frames
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DOI：
10.1137/21m1435446
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发表时间：
2021-09
期刊：
影响因子：
--
通讯作者：
R. Balan;Chris B. Dock
中科院分区：
文献类型：
--
作者：
R. Balan;Chris B. Dock
经典的相位恢复问题出现在从语音识别到X射线结晶学和量子状态层析成像的各种环境中。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。我们给出了拟线性分析映射$\beta的可计算的全局稳定界，并根据键空间的微分几何为理解相关问题提供了一条途径。特别地，我们证明了低秩半正定矩阵的惠特尼分层，这允许我们对全局稳定界的计算进行“分层”。我们证明了对于不纯状态情况，非线性分析映射关于某些自然距离度量不能得到这样的全局稳定界。最后，我们对$\beta$分析映射的全局下Lipschitz常数的计算为框架是广义相位可恢复提供了新的条件。
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 $\beta$ 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 $\alpha$ with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the $\beta$ analysis map provides novel conditions for a frame to be generalized phase retrievable.