论文标题
稳定的阶段检索和框架的扰动
Stable phase retrieval and perturbations of frames
论文作者
论文摘要
trage $（x_j）_ {j \ in J} $ for a Hilbert space $ h $据说可以进行阶段检索，如果对于所有不同的矢量$ x，y \ in h $ in h h $ a框架系数的幅度$（| \ langle x，x_jj \ rangle |） x_j \ rangle |）_ {j \ in J} $区分$ x $与$ y $（最多达到单模型的标量）。如果从框架系数的大小中恢复任何向量$ x \，则可以进行相位检索的框架可以进行$ c $稳定的相位检索。众所周知，如果帧进行稳定的相试验，那么框架向量的任何足够小的扰动都将进行稳定的相位检索，尽管稳定性常数稍差。我们提供了有关相位检索的稳定性常数如何受到框架向量的少量扰动的影响的新定量界限。这些界限很重要，因为它们独立于希尔伯特空间的维度和框架中的向量数。
A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ is said to do phase retrieval if for all distinct vectors $x,y\in H$ the magnitude of the frame coefficients $(|\langle x, x_j\rangle|)_{j\in J}$ and $(|\langle y, x_j\rangle|)_{j\in J}$ distinguish $x$ from $y$ (up to a unimodular scalar). A frame which does phase retrieval is said to do $C$-stable phase retrieval if the recovery of any vector $x\in H$ from the magnitude of the frame coefficients is $C$-Lipschitz. It is known that if a frame does stable phase retrieval then any sufficiently small perturbation of the frame vectors will do stable phase retrieval, though with a slightly worse stability constant. We provide new quantitative bounds on how the stability constant for phase retrieval is affected by a small perturbation of the frame vectors. These bounds are significant in that they are independent of the dimension of the Hilbert space and the number of vectors in the frame.
