PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

Candès, Emmanuel J.; Strohmer, Thomas; Voroninski, Vladislav

doi:10.1002/cpa.21432

Cited by 1,114 publications

(1,276 citation statements)

References 21 publications

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“…Recently, this family of problems has attracted a lot of attention in the mathematical community (see e.g. 2,3,14 or the research blog 10 ). A typical such problem is the phase retrieval problem in optics where one wants to reconstruct a compactly supported function ϕ from the modulus of its Fourier transform |F [ϕ]|.…”

Section: Introductionmentioning

confidence: 99%

Zak transform and non-uniqueness in an extension of Pauli’s phase retrieval problem

Andreys

Jaming

2016

Anal Math

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The aim of this paper is to pursue the investigation of the phase retrieval problem for the fractional Fourier transform F α started by the second author. We here extend a method of A.E.J.M Janssen to show that there is a countable set Q such that for every finite subset A ⊂ Q, there exist two functions f, g not multiple of one an other such that |F α f | = |F α g| for every α ∈ A. Equivalently, in quantum mechanics, this result reformulates as follows: if Q α = Q cos α + P sin α (Q, P be the position and momentum observables), then {Q α , α ∈ A} is not informationally complete with respect to pure states. This is done by constructing two functions ϕ, ψ such that F α ϕ and F α ψ have disjoint support for each α ∈ A. To do so, we establish a link between F α [f ], α ∈ Q and the Zak transform Z[f ] generalizing the well known marginal properties of Z.

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Section: Introductionmentioning

confidence: 99%

Zak transform and non-uniqueness in an extension of Pauli’s phase retrieval problem

Andreys

Jaming

2016

Anal Math

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“…As shown in [29], [31], this problem can be solved by convex programming via a low rank matrix completion formulation. As proven in [32], with high probability, this formulation is robust to measurement noise, and in fact (w.h.p.) yields the unique solution in the noise-free case, see also [33].…”

Section: ) Direct Solutionmentioning

confidence: 79%

Vectorial Phase Retrieval of 1-D Signals

Raz

Dudovich

Nadler

2013

IEEE Trans. Signal Process.

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Abstract-Reconstruction of signals from measurements of their spectral intensities, also known as the phase retrieval problem, is of fundamental importance in many scientific fields. In this paper we present a novel framework, denoted as vectorial phase retrieval, for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference. We show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal. First, we prove that for compactly supported signals, in the absence of measurement noise, this new setup admits a unique solution. Next, we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it. Finally, we illustrate via simulations, that our algorithm can accurately reconstruct signals even at considerable noise levels.Index Terms-Convex relaxation, 1-D phase retrieval, signal recovery from modulus Fourier measurements, statistical model selection.

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“…We now derive the first order optimality conditions −A * λ − I ∈ ∂ı K (M 0 ) for problem (9) by writing down the Lagrangian dual function for this problem, and by finding a dual vectorλ such that 0 ∈ ∂L(M 0 ,λ). Introducing multipliers for each of the polynomial constraints, the Lagrangian dual function can be written as…”

Section: Main Results and Mathematical Argumentmentioning

confidence: 99%

Rank-one matrix completion is solved by the sum-of-squares relaxation of order two

Cosse¹,

Demanet

2015

2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

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Abstract-This note studies the problem of nonsymmetric rank-one matrix completion. We show that in every instance where the problem has a unique solution, one can recover the original matrix through the second round of the sum-ofsquares/Lasserre hierarchy with minimization of the trace of the moments matrix. Our proof system is based on iteratively building a sum of N − 1 linearly independent squares, where N is the number of monomials of degree at most two, corresponding to the canonical basis (z α − z α 0 ) 2 . Those squares are constructed from the ideal I generated by the constraints and the monomials provided by the minimization of the trace. I. NON SYMMETRIC MATRIX COMPLETIONThis paper introduces a deterministic recovery result for non symmetric rank-1 matrix completion by using the Lasserre hierarchy of semidefinite programming relaxations. To our knowledge, the closest result in the current literature is [1] where the authors use the same hierarchy and certify recovery in the case of tensor decomposition. Our paper also shares its deterministic nature with [2] where the authors derive recovery from the spectral properties of a graph Laplacian. Finally, this paper can also be related to [3] in which the authors study noisy tensor completion and use the sixth round of the Lasserre hierarchy to derive probabilistic recovery guarantees.We will use M(r; m × n) to denote the set of matrices of rank r. This set is an algebraic determinantal variety that can be completely characterized through the vanishing of the (r + 1)-minors. This determinantal variety has dimension (m+n−r)r.The general nonsymmetric rank-1 matrix completion problem consists in recovering an unknown matrix X ∈ M(1; m× n) such that X = xy T , given a fixed subset of its entries [4], find X subject to rank(X) = 1As a slight abuse, we also speak of constraints X ij = A ij as belonging to Ω. In relation to problem (1), we introduce the mapping R Ω : R m×n → R |Ω| that corresponds to extracting the observed entries of the matrix. We let R 1 Ω denote the restriction of R Ω to matrices of rank-1, i.e R 1 Ω : M(1; m × n) → R |Ω| . Invertibility of this restriction R 1 Ω is a natural question. In other words, when can one uniquely recover the matrix X from the knowledge of R Ω (X) and the fact that X has rank 1 ?In particular, this paper considers the completion problem on M * (1, m×n), where M * (1, m×n) denotes the restriction of M(1; m × n) to matrices for which none of the entries are zero. The reason for this is that if a rank-1 matrix has a zero element, then the corresponding row or column will be zero, and it is easy to see that the completion problem will generically lack injectivity.Respectively denote by V 1 , V 2 the row and column indices of X. We consider the bipartite graph G(V 1 , V 2 , E) associated to problem (1), where the set of edges in the graph is defined by (i, j) ∈ E iff (i, j) ∈ Ω. The conditions for the recovery of the matrix X from the set Ω are related to the properties of this bipartite graph. In particular, we have the f...

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PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

Cited by 1,114 publications

References 21 publications

Zak transform and non-uniqueness in an extension of Pauli’s phase retrieval problem

Zak transform and non-uniqueness in an extension of Pauli’s phase retrieval problem

Vectorial Phase Retrieval of 1-D Signals

Rank-one matrix completion is solved by the sum-of-squares relaxation of order two

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