Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns

Candès, Emmanuel J.; Li, Xiaodong

doi:10.1007/s10208-013-9162-z

Cited by 272 publications

(403 citation statements)

References 6 publications

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“…In a separate paper, [14], the authors developed further a similar algorithm in the case of windowed DFT. The original requirement of m = O(n log(n)) vectors has been relaxed to m = O(n) in [15]. Similar convex optimization solutions have been proposed by the authors of [20] and [34].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Bălan

2015

Found Comput Math

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Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well as an iterative algorithm that finds the least-square solution which is robust in the presence of noise. We analyze its numerical performance by comparing it to the Cramer-Rao lower bound.

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Section: Introductionmentioning

confidence: 99%

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Bălan

2015

Found Comput Math

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“…The first convex relaxation of the phase retrieval problem has been introduced by Candes et al [12]- [14]. He observed that the non-convex measurements y on vectors x become linear measurements on matrices X = xx H where (.)…”

Section: A Phaseliftmentioning

confidence: 99%

“…The equalities in (5) no longer hold in presence of noise. The following formulation has then been proposed in [14]:…”

Section: A Phaseliftmentioning

confidence: 99%

“…It means that under appropriate conditions, the difficult problem (2) is equivalent to the convex program (6) or (11) in the sense that they have the same unique solution. If the number M of equations |a m T x| = y m are sufficiently randomized and at least on the order of N , it has been demonstrated in [14] that the solution to the convex relaxation (6) is exact (i.e. it is the same as the one of (2)) with a probability approaching one exponentially, as M is increased.…”

Section: ) Sampling (M/n )mentioning

confidence: 99%

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Excitation Retrieval of Microwave Linear Arrays From Phaseless Far-Field Data

Fuchs

Coq

2015

IEEE Trans. Antennas Propagat.

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Abstract-A methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed. The approach combines tools from convex optimization (to solve the phase retrieval problem) and a simple measurement procedure (to mitigate the non uniqueness of the solution). Numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed. They show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem. In addition, the proposed approach, that only calls for readily available routines, is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases.

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“…Up to now, there are two main categories of phase retrieval approaches in existing approaches, namely, semi-definite programming-based (SDP-based) approaches [7][8][9][10][11] and iterative projection approaches (Fienup-type methods) [12][13][14][15][16][17][18]. SDP-based approaches are not suitable for largescale problems, so we do not discuss them in the paper.…”

Section: Introductionmentioning

confidence: 99%

Iterative projection approach for phase retrieval of semi-sparse wave field

Fan

Wan

Wen

et al. 2014

EURASIP J. Adv. Signal Process.

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In the paper, we consider the problem of two-dimensional (2D) phase retrieval, which recovers a 2D complex-valued wave field from magnitudes of both wave field and its Fourier transform. Due to the absence of the phase measurements, prior information on wave field is needed in order to recover phase, which is feasible when the phases of the wave field are sparse. In this paper, we improve the phase retrieval accuracy by incorporating phase sparse constraint of wave field. As a sequel to previous iterative projection approaches, iterative projection approaches with phase sparse constraint are realized based on 'soft thresholding'. It has superior performances in terms of convergence, residual error, noise stability, and suitability in large-scale phase retrieval problems. Numerical experiments illustrate that the proposed approach outperforms existing iterative projection approaches.

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Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns

Cited by 272 publications

References 6 publications

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Excitation Retrieval of Microwave Linear Arrays From Phaseless Far-Field Data

Iterative projection approach for phase retrieval of semi-sparse wave field

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