Real phase retrieval via finite real measurements

Let $d\ge 1$, $m\ge 0$, and $\mathrm{\Phi }=(a_i{)}_{i\in [m]}$ be a finite family in ${\mathbb{R}}^d$, allowing zero measurement vectors and not assuming that $\mathrm{\Phi }$ spans. Here $[m]=\{1,\dots ,m\}$ and $[0]=\mathrm{\varnothing }$; measurement labels are one-based, while coordinates in ${\mathbb{R}}^d$ use the library's zero-based convention $k<d$. Write ${\mathbb{R}}^{[m]}=\{y:[m]\to \mathbb{R}\}$. Use the Euclidean inner product $⟨u,v⟩={\sum }_{k<d}{u}_k{v}_k$ and norm $\mathrm{\parallel }u{\mathrm{\parallel }}_2=\sqrt{⟨u,u⟩}$. Define the signal equivalence relation $x\sim z$ by $z=x$ or $z=-x$, let $Q_d={\mathbb{R}}^d\mathrm{/}\sim$, and write $q(x)=[x]$. Its quotient distance is $d_{\pm }([x],[z]):=\min \{\mathrm{\parallel }x-z{\mathrm{\parallel }}_2,\mathrm{\parallel }x+z{\mathrm{\parallel }}_2\}\mathrm{.}$

Use data space $Y_m=\{b:[m]\to [0,\mathrm{\infty })\}$ with metric ${\rho }_m(b,c)=\{\begin{array}{ll}\sqrt{{\sum }_{k<m}(b_{k+1}-c_{k+1}{)}^2},& m\ge 1,\\ 0,& m=0.\end{array}$ The fixed relabeling $i\mapsto i-1$ identifies $[m]$-indexed data with the library's zero-based ${\mathbb{R}}^m$; $Y_0$ is a singleton. Define the real analysis map $T_{\mathrm{\Phi }}:{\mathbb{R}}^d\to {\mathbb{R}}^{[m]}$ by $T_{\mathrm{\Phi }}x=(⟨a_i,x⟩{)}_{i\in [m]}\mathrm{.}$ The amplitude map is ${\mathcal{M}}_{\mathrm{\Phi }}([x])=(\mathrm{\mid }⟨a_i,x⟩\mathrm{\mid }{)}_{i\in [m]}$, while the intensity map is ${\mathcal{I}}_{\mathrm{\Phi }}([x])=(⟨a_i,x{⟩}^2{)}_{i\in [m]}\in Y_m\mathrm{.}$ A finite real measurement family performs phase retrieval when ${\mathcal{I}}_{\mathrm{\Phi }}:Q_d\to Y_m$ is injective, equivalently when the induced amplitude map is injective. The quotient distance, both induced maps, and the data metric are well defined as verified below.

The diagram records the identity ${\mathcal{I}}_{\mathrm{\Phi }}\circ q=S\circ T_{\mathrm{\Phi }}$, where $S(y)=(y_i^2{)}_{i\in [m]}$ is coordinatewise squaring. It distinguishes signed linear measurements, amplitudes, and intensities; no proof relies on the diagram.

diagram failed to render:
\mathbb R^d \arrow[r,"T_\Phi"] \arrow[d,"q"'] & \mathbb R^{[m]} \arrow[d,"S"] \\
Q_d \arrow[r,"\mathcal I_\Phi"'] & [0,\infty)^m

Forward maps and target-class spaces

Identifiability modulo a target equivalence

Minimum of a finite set of real numbers

The real order used to attain this two-point minimum

Euclidean inner product and norm

Euclidean metrics on finite real coordinate spaces

Radu Balan, Pete Casazza, and Dan Edidin, The real finite-frame measurement model and injectivity, §2

Philipp Grohs, Sarah Koppensteiner, and Martin Rathmair, abstract phaseless maps and quotient classes, §§1–2.1

Radu Balan, real phase retrieval and complement property, §3.2