Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns

2026-09-25

AI Summary

This study addresses the challenge of stability analysis in phase retrieval arising from dependencies among Fourier measurements in coded diffraction patterns by establishing a unified lower isometry property framework. Methodologically, it introduces a row-subset operator norm bounding technique that preserves tangent injectivity while adaptively removing dependent rows. By integrating random masks with PhaseLift convex programming and non-convex factorization losses, the theoretical analysis is completed via matrix concentration inequalities. The primary contribution demonstrates that merely O(log n) masks suffice to achieve Gaussian-type stable recovery through convex optimization and to certify benign geometry for the non-convex landscape. These results rigorously guarantee noise robustness as well as global optimal convergence in the noiseless setting.

Abstract

Coded diffraction patterns (CDPs) provide a structured and physically relevant model for phase retrieval, but the dependence among Fourier measurements generated by a common mask makes sharp stability analysis challenging. For a fixed unit-norm signal $\boldsymbol{x}_\star \in \mathbb{C}^n$, let $\boldsymbol{X}_\star=\boldsymbol{x}_\star\boldsymbol{x}_\star^*$, and let $\mathcal A$ be the lifted CDP measurement operator. We prove that, with $L=O(\log n)$ random masks, the following uniform lower isometry holds with high probability: $\|\boldsymbol{X}-\boldsymbol{X}_\star\|_F \lesssim \log^2(2n) \frac{\|\mathcal A(\boldsymbol{X}-\boldsymbol{X}_\star)\|_2}{\sqrt{nL}}, \boldsymbol{X}\succeq\boldsymbol{0},$ from which we derive two consequences. First, for $\boldsymbol{y}=\mathcal A(\boldsymbol{X}_\star)+\boldsymbol{e}$, PhaseLift-type convex programs achieve the Gaussian-type stable recovery bound $\|\widehat{\boldsymbol{X}}-\boldsymbol{X}_\star\|_F \lesssim \frac{\log^2(2n)}{\sqrt{nL}}\|\boldsymbol{e}\|_2.$ Second, in the noiseless case, the nonconvex factorized loss has a benign landscape when the factor width satisfies $r = O(\log^5(2n))$: every second-order critical point $\boldsymbol{V}\in\mathbb C^{n\times r}$ satisfies $\boldsymbol{V}\boldsymbol{V}^*=\boldsymbol{X}_\star$. The key ingredient is a uniform operator-norm bound over row subsets of the dependent CDP measurement matrix, which permits the removal of a controlled set of adaptively selected rows while preserving tangent injectivity.

Problem

• Phase Retrieval
• Coded Diffraction Patterns
• Stable Recovery
• Overparameterization
• Benign Landscape

Innovation

• Phase Retrieval
• Coded Diffraction Patterns
• Stable Recovery
• Benign Landscape
• Overparameterization