Phase Retrieval With Random Gaussian Sensing Vectors by Alternating Projections

Waldspurger, Irène

doi:10.1109/tit.2018.2800663

Cited by 86 publications

(108 citation statements)

References 26 publications

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“…The analysis of bounding P x ⊥ d is similar to that of [11]. Proof By the definition (6.12) of d(z), notice that…”

Section: Boundingmentioning

confidence: 91%

“…ADM iteration. ADM is a classical method for solving phase retrieval problems [11,26,33], which can be considered as a heuristic method for solving the following nonconvex problem min z∈C n ,|u|=1…”

Section: Proof Sketch Of Iterative Contractionmentioning

confidence: 99%

“…Our proof is based on ideas from decoupling theory [12], the suprema of chaos processes and restricted isometry property of random circulant matrices [13,14], and is also inspired by a new iterative analysis of alternating minimization methods [11]. Our analysis draws connections between the convergence properties of gradient descent and the classical alternating direction method.…”

Section: Introductionmentioning

confidence: 99%

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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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“…The analysis of bounding P x ⊥ d is similar to that of [11]. Proof By the definition (6.12) of d(z), notice that…”

Section: Boundingmentioning

confidence: 91%

Section: Proof Sketch Of Iterative Contractionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convolutional Phase Retrieval via Gradient Descent

Zhang

Eldar

et al. 2020

IEEE Trans. Inform. Theory

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show abstract

“…We show that, with a random initialization, the classical algorithm of alternating minimization succeeds with high probability as n, m → ∞ when m/log 3 m ≥ M n 3/2 log 1/2 n for some M > 0. This is a step toward proving the conjecture in [27], which conjectures that the algorithm succeeds when m = O(n). The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.…”

mentioning

confidence: 94%

“…The most widely used method is perhaps the alternate minimization (Gerchberg-Saxton) algorithm and its variants [14], [12], [13], that is based on alternating projections onto nonconvex sets [2]. As a result, in some literature it is also called the alternating projection method [27]. This method is very simple T. Zhang to implement and is parameter-free.…”

Section: Introductionmentioning

confidence: 99%

Phase Retrieval by Alternating Minimization With Random Initialization

Zhang

2020

IEEE Trans. Inform. Theory

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We consider a phase retrieval problem, where the goal is to reconstruct a n-dimensional complex vector from its phaseless scalar products with m sensing vectors, independently sampled from complex normal distributions. We show that, with a random initialization, the classical algorithm of alternating minimization succeeds with high probability as n, m → ∞ when m/log 3 m ≥ M n 3/2 log 1/2 n for some M > 0. This is a step toward proving the conjecture in [27], which conjectures that the algorithm succeeds when m = O(n). The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.

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Compressive phase retrieval: Optimal sample complexity with deep generative priors

Hand,

Leong,

Voroninski

2023

Comm Pure Appl Math

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Advances in compressive sensing (CS) provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that succeed with optimal sample complexity? Meanwhile, progress in machine learning has led to the development of new data‐driven signal priors in the form of generative models, which can outperform sparsity priors with significantly fewer measurements. In this work, we resolve the open problem in compressive phase retrieval and demonstrate that generative priors can lead to a fundamental advance by permitting optimal sample complexity by a tractable algorithm. We additionally provide empirics showing that exploiting generative priors in phase retrieval can significantly outperform sparsity priors. These results provide support for generative priors as a new paradigm for signal recovery in a variety of contexts, both empirically and theoretically. The strengths of this paradigm are that (1) generative priors can represent some classes of natural signals more concisely than sparsity priors, (2) generative priors allow for direct optimization over the natural signal manifold, which is intractable under sparsity priors, and (3) the resulting non‐convex optimization problems with generative priors can admit benign optimization landscapes at optimal sample complexity, perhaps surprisingly, even in cases of nonlinear measurements.

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Phase Retrieval With Random Gaussian Sensing Vectors by Alternating Projections

Cited by 86 publications

References 26 publications

Convolutional Phase Retrieval via Gradient Descent

Convolutional Phase Retrieval via Gradient Descent

Phase Retrieval by Alternating Minimization With Random Initialization

Compressive phase retrieval: Optimal sample complexity with deep generative priors

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