基于仿射变换范数的相位恢复
Phase retrieval from the norms of affine transformations
摘要：
本文考虑了从仿射测量中恢复广义相位的问题。这个问题的目的是恢复信号${\mathbf x} \in {\mathbb F}^d$从仿射测量$y_j=\norm{M_j^*\vx +{\mathbb b}_j}^2,\; j=1,\ldots,m,$，其中$M_j \in {\mathbb F}^{d\times r}, {\mathbf b}_j\in {\mathbb F}^{r}, {\mathbb F}\in \{{\mathbb R},{\mathbb C}\}$，我们称之为{ \em广义仿射相位恢复}。提出了广义仿射相位恢复的框架，给出了$\{(M_j,{\mathbf b}_j)\}_{j=1}^m$具有广义仿射相位恢复性质的充分必要条件。我们还建立了广义仿射相位恢复的最小测量数的结果。特别地，我们证明了如果$\{(M_j,{\mathbf b}_j)\}_{j=1}^m \subset {\mathbb F}^{d\times r}\times {\mathbb F}^{r}$具有广义仿射相位恢复特性，则对于${\mathbb F}={\mathbb R}$的$m\geq d+\floor{d/r}$（对于${\mathbb F}={\mathbb C}$的$m\geq 2d+\floor{d/r}$）。我们还表明，该界是紧提供$r\mid d$。这些结果意味着可以通过提高$r$（即$M_j$的等级）来减少测量次数。这突出了广义仿射相位恢复和广义相位恢复之间的显著差异。此外，使用代数几何的工具，我们表明，$m\geq 2d$（resp. $m\geq 4d-1$）通用测量${\mathcal A}=\{(M_j,b_j)\}_{j=1}^m$具有用于${\mathbb F}={\mathbb R}$的广义相位恢复特性（分别${\mathbb F}={\mathbb C}$）。
英文摘要：
In this paper, we consider the generalized phase retrieval from affine measurements. This problem aims to recover signals ${\mathbf x} \in {\mathbb F}^d$ from the affine measurements $y_j=\norm{M_j^*\vx +{\mathbb b}_j}^2,\; j=1,\ldots,m,$ where $M_j \in {\mathbb F}^{d\times r}, {\mathbf b}_j\in {\mathbb F}^{r}, {\mathbb F}\in \{{\mathbb R},{\mathbb C}\}$ and we call it as {\em generalized affine phase retrieval}. We develop a framework for generalized affine phase retrieval with presenting necessary and sufficient conditions for $\{(M_j,{\mathbf b}_j)\}_{j=1}^m$ having generalized affine phase retrieval property. We also establish results on minimal measurement number for generalized affine phase retrieval. Particularly, we show if $\{(M_j,{\mathbf b}_j)\}_{j=1}^m \subset {\mathbb F}^{d\times r}\times {\mathbb F}^{r}$ has generalized affine phase retrieval property, then $m\geq d+\floor{d/r}$ for ${\mathbb F}={\mathbb R}$ ($m\geq 2d+\floor{d/r}$ for ${\mathbb F}={\mathbb C}$ ). We also show that the bound is tight provided $r\mid d$. These results imply that one can reduce the measurement number by raising $r$, i.e. the rank of $M_j$. This highlights a notable difference between generalized affine phase retrieval and generalized phase retrieval. Furthermore, using tools of algebraic geometry, we show that $m\geq 2d$ (resp. $m\geq 4d-1$) generic measurements ${\mathcal A}=\{(M_j,b_j)\}_{j=1}^m$ have the generalized phase retrieval property for ${\mathbb F}={\mathbb R}$ (resp. ${\mathbb F}={\mathbb C}$).
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