The Condition Number in Phase Retrieval from Intensity Measurements

TL;DR

This paper analyzes the stability of phase retrieval by establishing universal lower bounds on the condition number of the nonlinear measurement map, revealing fundamental limits and optimal sensing matrices for stable reconstruction.

Contribution

It provides the first explicit uniform lower bounds on the condition number in phase retrieval, demonstrating their tightness and identifying optimal sensing matrices.

Findings

Lower bounds: /2 for real  and  for complex cases.

Harmonic frames and Gaussian matrices asymptotically attain bounds.

Harmonic frame  achieves optimal stability for real 2D phase retrieval.

Abstract

This paper investigates the stability of phase retrieval by analyzing the condition number of the nonlinear map $\Psi_{\boldsymbol{A}}(\boldsymbol{x}) = \bigl(\lvert \langle {\boldsymbol{a}}_j, \boldsymbol{x} \rangle \rvert^2 \bigr)_{1 \le j \le m}$, where $\boldsymbol{a}_j \in \mathbb{H}^n$ are known sensing vectors with $\mathbb{H} \in \{\mathbb{R}, \mathbb{C}\}$. For each $p \ge 1$, we define the condition number $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$ as the ratio of optimal upper and lower Lipschitz constants of $\Psi_{\boldsymbol{A}}$ measured in the $\ell_p$ norm, with respect to the metric $\mathrm {dist}_\mathbb{H}\left(\boldsymbol{x}, \boldsymbol{y}\right) = \|\boldsymbol{x} \boldsymbol{x}^\ast - \boldsymbol{y} \boldsymbol{y}^\ast\|_*$. We establish universal lower bounds on $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$ for any sensing matrix $\boldsymbol{A} \in \mathbb{H}^{m \times d}$, proving that $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_1} \ge \pi/2$ and $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_2} \ge \sqrt{3}$ in the real case $(\mathbb{H} = \mathbb{R})$, and $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p} \ge 2$ 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 $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ and Gaussian random matrices $\boldsymbol{A} \in \mathbb{H}^{m \times d}$ asymptotically attain them. Notably, the harmonic frame $\boldsymbol{E}_m \in \mathbb{R}^{m \times 2}$ achieves the optimal lower bound $\sqrt{3}$ for all $m \ge 3$ when $p=2$, thus serving as an optimal sensing matrix within $\boldsymbol{A} \in \mathbb{R}^{m \times 2}$. Our results provide the first explicit uniform lower bounds on $\beta_{\Psi_{\boldsymbol{A}}}^{\ell_p}$ and offer insights into the fundamental stability limits of phase retrieval.