The Condition Number in Phase Retrieval from Intensity Measurements

This paper investigates the stability of phase retrieval by analyzing the condition number of the nonlinear map ${\mathrm{\Psi }}_{\mathit{A}}(\mathit{x})=(\mid ⟨{\mathit{a}}_j,\mathit{x}⟩{\mid }^2{)}_{1\le j\le m}$, where ${\mathit{a}}_j\in {\mathbb{H}}^n$ are known sensing vectors with $\mathbb{H}\in \{\mathbb{R},\mathbb{C}\}$. For each $p\ge 1$, we define the condition number ${\beta }_{{\mathrm{\Psi }}_{\mathit{A}}}^{{\mathrm{\ell }}_p}$ as the ratio of optimal upper and lower Lipschitz constants of ${\mathrm{\Psi }}_{\mathit{A}}$ measured in the ${\mathrm{\ell }}_p$ norm, with respect to the metric ${\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}_{\mathbb{H}}(\mathit{x},\mathit{y})=\mathrm{\parallel }\mathit{x}{\mathit{x}}^{\ast }-\mathit{y}{\mathit{y}}^{\ast }{\mathrm{\parallel }}_{\ast }$. We establish universal lower bounds on ${\beta }_{{\mathrm{\Psi }}_{\mathit{A}}}^{{\mathrm{\ell }}_p}$ for any sensing matrix $\mathit{A}\in {\mathbb{H}}^{m\times d}$, proving that ${\beta }_{{\mathrm{\Psi }}_{\mathit{A}}}^{{\mathrm{\ell }}_1}\ge \pi \mathrm{/}2$ and ${\beta }_{{\mathrm{\Psi }}_{\mathit{A}}}^{{\mathrm{\ell }}_2}\ge \sqrt{3}$ in the real case $(\mathbb{H}=\mathbb{R})$, and ${\beta }_{{\mathrm{\Psi }}_{\mathit{A}}}^{{\mathrm{\ell }}_p}\ge 2$ for $p=1,2$ in the complex case $(\mathbb{H}=\mathbb{C})$. These bounds are shown to be asymptotically tight: both a deterministic harmonic frame ${\mathit{E}}_m\in {\mathbb{R}}^{m\times 2}$ and Gaussian random matrices $\mathit{A}\in {\mathbb{H}}^{m\times d}$ asymptotically attain them. Notably, the harmonic frame ${\mathit{E}}_m\in {\mathbb{R}}^{m\times 2}$ achieves the optimal lower bound $\sqrt{3}$ for all $m\ge 3$ when $p=2$, thus serving as an optimal sensing matrix within $\mathit{A}\in {\mathbb{R}}^{m\times 2}$. Our results provide the first explicit uniform lower bounds on ${\beta }_{{\mathrm{\Psi }}_{\mathit{A}}}^{{\mathrm{\ell }}_p}$ and offer insights into the fundamental stability limits of phase retrieval.