Phase retrieval from the norms of affine transformations

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 $yj=\norm{Mj^*\vx +{\mathbb b}j}^2,\; j=1,\ldots,m,$ where $Mj \in {\mathbb F}^{d\times r}, {\mathbf b}j\in {\mathbb F}^{r}, {\mathbb F}\in {{\mathbb R},{\mathbb C}}$ 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 ${(Mj,{\mathbf b}j)}{j=1}^m$ having generalized affine phase retrieval property. We also establish results on minimal measurement number for generalized affine phase retrieval. Particularly, we show if ${(Mj,{\mathbf b}j)}{j=1}^m \subset {\mathbb F}^{d\times r}\times {\mathbb F}^{r}$ 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 $Mj$. 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 ${\mathcal A}={(Mj,bj)}_{j=1}^m$ have the generalized phase retrieval property for ${\mathbb F}={\mathbb R}$ (resp. ${\mathbb F}={\mathbb C}$).