The Condition Number in Phase Retrieval from Intensity Measurements

2025-06-27

AI Summary

This work investigates the fundamental stability limits of phase retrieval under intensity-only measurements, focusing on characterizing the condition number of the associated nonlinear mapping Ψₐ(x). We define the condition number as the ratio of Lipschitz constants and establish, for the first time, universal lower bounds on it under ℓₚ norms. Tight lower bounds are derived separately for real and complex settings. Theoretically, we prove that harmonic frames achieve the optimal bound in two-dimensional real space; moreover, Gaussian random matrices and specific harmonic frames asymptotically attain these bounds, confirming their tightness and achievability. Our results provide the first unified, tight, and achievable stability benchmark for phase retrieval, revealing intrinsic limitations in measurement system design and identifying optimal frame structures.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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

Problem

• Analyzes stability of phase retrieval via condition number
• Establishes universal lower bounds for condition numbers
• Identifies optimal sensing matrices for phase retrieval

Innovation

• Analyzes condition number for phase retrieval stability
• Establishes universal lower bounds for sensing matrices
• Demonstrates asymptotic tightness with harmonic frames