Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns
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 x⋆ ∈ ℂn, let X⋆=x⋆x⋆*, and let 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: |X−X⋆|F ≲ log2(2n) (|𝒜(X−X⋆)|2)/√(nL), X⪰0, from which we derive two consequences. First, for y=𝒜(X⋆)+e, PhaseLift-type convex programs achieve the Gaussian-type stable recovery bound |X̂−X⋆|F ≲ log2(2n)/√(nL)|e|2. Second, in the noiseless case, the nonconvex factorized loss has a benign landscape when the factor width satisfies r = O(log5(2n)): every second-order critical point V∈ℂn× r satisfies VV*=X⋆. 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.
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BibTeX
@misc{cai2026stable,
title = {{Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns}},
author = {Jian-Feng Cai and Zhibo Jin and Tong Wu and Ruizhe Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30825},
eprint = {2609.30825},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30825},
}
RIS
TY - UNPB
TI - Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns
AU - Jian-Feng Cai
AU - Zhibo Jin
AU - Tong Wu
AU - Ruizhe Xia
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30825
UR - https://arxiv.org/abs/2609.30825
AB - 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 x_⋆ ∈ ℂ^n, let X_⋆=x_⋆x_⋆^*, and let 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: |X−X_⋆|_F ≲ log^2(2n) (|𝒜(X−X_⋆)|_2)/√(nL), X⪰0, from which we derive two consequences. First, for y=𝒜(X_⋆)+e, PhaseLift-type convex programs achieve the Gaussian-type stable recovery bound |X̂−X_⋆|_F ≲ log^2(2n)/√(nL)|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 V∈ℂ^n× r satisfies VV^*=X_⋆. 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.
ER -