Phase retrieval from the norms of affine transformations

Abstract: In this paper, we consider the generalized phase retrieval from affine measurements. This problem aims to recover signals ${\mathbf x} \in {\mathbb F}<sup>d$ from the affine measurements $y_j=\norm{M_j^*\vx +{\mathbb b}<em>j}^2,\; j=1,\ldots,m,$ 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 ${(M_j,{\mathbf b}_j)}</em>{j=1}<sup>m$ having generalized affine phase retrieval property. We also establish results on minimal measurement number for generalized affine phase retrieval. Particularly, we show if $${(M_j,{\mathbf b}<em>j)}</em>{j=1}^m \subset {\mathbb F}^{d\times r}\times {\mathbb F}<sup>{r}$$ 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 ${\mathcal A}={(M_j,b_j)}_{j=1}<sup>m$ have the generalized phase retrieval property for ${\mathbb F}={\mathbb R}$ (resp. ${\mathbb F}={\mathbb C}$).