PhaseMax: Convex Phase Retrieval via Basis Pursuit

Goldstein, Tom; Studer, Christoph

doi:10.1109/tit.2018.2800768

Cited by 244 publications

(163 citation statements)

References 60 publications

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“…The Gerchberg-Saxton Method of Alternating Projections. For a matrix that does phase retrieval on C K , there are many reconstruction techniques: frame methods [6,5]; convex optimization [10,16]; and the Kaczmarz method [36,31] to name only a few. However, for a matrix V that does conjugate phase retrieval on C K (but not phase retrieval), there is no known proven method of reconstruction.…”

Section: Numerical Methods and Experimentsmentioning

confidence: 99%

Conjugate Phase Retrieval in Paley-Wiener Space

Lai

Littmann

Weber

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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We consider the problem of conjugate phase retrieval in Paley-Wiener space P W π . The goal of conjugate phase retrieval is to recover a signal f from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning f (t) and f (t) are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in P W π by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in P W π by sampling both f and f only on the real line. Moreover, we demonstrate experimentally that the Gerchberg-Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate.

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Section: Numerical Methods and Experimentsmentioning

confidence: 99%

Conjugate Phase Retrieval in Paley-Wiener Space

Lai

Littmann

Weber

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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“…However, the high computational cost of SDP limits their practicality. Quite recently, [30][31][32] reveal that the problem can also be solved in the natural parameter space via linear programming.…”

Section: Comparison With Literaturementioning

confidence: 99%

Convolutional Phase Retrieval via Gradient Descent

Zhang

Eldar

et al. 2020

IEEE Trans. Inform. Theory

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We study the convolutional phase retrieval problem, of recovering an unknown signal x ∈ C n from m measurements consisting of the magnitude of its cyclic convolution with a given kernel a ∈ C m . This model is motivated by applications such as channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when a is random and the number of observations m is sufficiently large, with high probability x can be efficiently recovered up to a global phase shift using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator. We overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis of alternating minimization methods. ). Most of the work has been conducted when QQ and YZ were with Department of Electrical Engineering and Data Science Institute at Columbia University. An extended abstract of the current work has appeared in NeurIPS'17 [1]. 1 The convolution can be written in matrix-vector form as a x = Caιn→mx, where Ca ∈ R m×m denotes the circulant matrix generated by a and ιn→m is a zero padding operator. In other words, a x = Ax with A ∈ C m×n being a matrix formed by the first n columns of Ca. arXiv:1712.00716v3 [stat.CO] 6 Oct 2019Structured random measurements. The study of structured random measurements in signal processing has quite a long history [44]. For compressed sensing [45], the work [46-48] studied random Fourier measurements, and later [13,14] proved similar results for partial random convolution measurements. However, the study of structured random measurements for phase retrieval is still quite limited. In particular, [49] and [50] studied t-designs and coded diffraction patterns (i.e., random masked Fourier measurements) using semidefinite programming. Recent work studied nonconvex optimization using coded diffraction patterns [34] and STFT measurements [51], both of which minimize a nonconvex objective similar to (1.5). These measurement models are motivated by different applications. For instance, coded diffraction is designed for imaging applications such as X-ray diffraction imaging, STFT can be applied to frequency resolved optical gating [52] and some speech processing tasks [53]. Both of the results show iterative contraction in a region that is at most O(1/ √ n)-close to the optimum. Unfortunately, for both results either the radius of the contraction region is not large enough for initialization to reach, or they require extra artificial technique such as resampling the data. In comparison, the contraction region we show for the random convolutional model is larger O(1/polylog(n)), which is achievable in the initialization stage via the spectral method. For a more detailed review of this subject, we refer the readers to Section 4 of [44]....

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“…where i=1, 2, K, n. In equation (14), U is the differential of gradient g in equation (12), so H is the differential of G, and in (16), g i represents the column vector of g . In fact, the problem of solving the objective function in equation (11) translates into an optimization problem.…”

Section: The New Proposed Rrcsfsl0 Algorithm and Its Stepsmentioning

confidence: 99%

“…The disadvantage of greedy algorithm is that it is sensitive to noise, and its computational complexity is increased with respect to signal sparse (k, the number of non-zero elements in x). Convex relaxation algorithm, such as Basis Pursuit (BP) [15,16], Least Absolute Shrinkage and Selection Operator (LASSO) [17], basis pursuit denoising (BPDN) [18,19], reconstructs the sparse signal by linear programming, but its computational complexity is increased . Non-convex relaxation algorithm reformulates equation (3) as [20,21] x y x argmin 1 2 4…”

Section: Introductionmentioning

confidence: 99%

A re-weighted smoothed L0 -norm regularized sparse reconstructed algorithm for linear inverse problems

Wang

Xiang

et al. 2019

J. Phys. Commun.

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This paper addresses the problems of sparse signal and image recovery using compressive sensing (CS), especially in the case of Gaussian noise. The main contribution of this paper is the proposal of the regularization re-weighted Composite Sine function smoothed L 0 -norm minimization (RRCSFSL0) algorithm where the Composite Sine function (CSF), the iteratively re-weighted scheme and the regularization mechanism represent the core of an approach to the solution of the problem. Compared with other state-of-the-art functions, the CSF we proposed can better approximate the L 0 -norm and improve the reconstruction accuracy, the new re-weighted scheme we adopted can promote sparsity and speed up convergence. Moreover, the use of the regularization mechanism makes the RRCSFSL0 algorithm more robust against noise. The performance of the proposed algorithm is verified via numerical experiments in the noise environment. Furthermore, experiments and comparisons demonstrate the superiority of the RRCSFSL0 algorithm in image restoration.

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PhaseMax: Convex Phase Retrieval via Basis Pursuit

Cited by 244 publications

References 60 publications

Conjugate Phase Retrieval in Paley-Wiener Space

Conjugate Phase Retrieval in Paley-Wiener Space

Convolutional Phase Retrieval via Gradient Descent

A re-weighted smoothed L0 -norm regularized sparse reconstructed algorithm for linear inverse problems

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