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Gabor Phase Retrieval via Semidefinite Programming.

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Title: Gabor Phase Retrieval via Semidefinite Programming.
Authors: Jaming, Philippe1 (AUTHOR) philippe.jaming@math.u-bordeaux.fr, Rathmair, Martin2 (AUTHOR) martin.rathmair@univie.ac.at
Source: Foundations of Computational Mathematics. Feb2026, Vol. 26 Issue 1, p245-311. 67p.
Subjects: Gabor transforms, Semidefinite programming, Inverse problems, Convex programming, Signal reconstruction, Mathematical complex analysis
Abstract: We consider the problem of reconstructing a function f ∈ L 2 (R) given phase-less samples of its Gabor transform, which is defined by G f (x , y) : = 2 1 4 ∫ R f (t) e - π (t - x) 2 e - 2 π i y t d t , (x , y) ∈ R 2. More precisely, given sampling positions Ω ⊆ R 2 the task is to reconstruct f (up to global phase) from measurements { | G f (ω) | : ω ∈ Ω } . This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime. Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion. As a byproduct, we also obtain improved truncation error for Gabor expensions with Gaussian generators. [ABSTRACT FROM AUTHOR]
Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: We consider the problem of reconstructing a function f ∈ L 2 (R) given phase-less samples of its Gabor transform, which is defined by G f (x , y) : = 2 1 4 ∫ R f (t) e - π (t - x) 2 e - 2 π i y t d t , (x , y) ∈ R 2. More precisely, given sampling positions Ω ⊆ R 2 the task is to reconstruct f (up to global phase) from measurements { | G f (ω) | : ω ∈ Ω } . This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime. Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion. As a byproduct, we also obtain improved truncation error for Gabor expensions with Gaussian generators. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10208-024-09683-6
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      – SubjectFull: Inverse problems
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              Text: Feb2026
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