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22
ROBUST SPARSE PHASE RETRIEVAL MADE EASY
"... Abstract. In this short note we propose a simple twostage sparse phase retrieval strategy that uses a nearoptimal number of measurements, and is both computationally efficient and robust to measurement noise. In addition, the proposed strategy is fairly general, allowing for a large number of new ..."
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Abstract. In this short note we propose a simple twostage sparse phase retrieval strategy that uses a nearoptimal number of measurements, and is both computationally efficient and robust to measurement noise. In addition, the proposed strategy is fairly general, allowing for a large number of new measurement constructions and recovery algorithms to be designed with minimal effort. 1.
FRAMES AND PHASELESS RECONSTRUCTION AMS SHORT COURSE: JOINT MATHEMATICS MEETINGS . . .
"... Frame design for phaseless reconstruction is now part of the broader problem of nonlinear recon struction and is an emerging topic in harmonic analysis. The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redunda ..."
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Frame design for phaseless reconstruction is now part of the broader problem of nonlinear recon struction and is an emerging topic in harmonic analysis. The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redundant system (frame), we want to reconstruct the unknown input. This problem has first occurred in Xray crystallography starting from the early 20th century. The same nonlinear reconstruction problem shows up in speech processing, particularly in speech recognition. In this lecture we shall cover existing analysis results as well as algorithms for signal recovery including: necessary and sufficient conditions for injectivity, Lipschitz bounds of the nonlinear map and its left inverses, stochastic performance bounds, convex relaxation algorithms for inversion, leastsquares inversion algorithms.
STABILITY OF FRAMES WHICH GIVE PHASE RETRIEVAL
"... In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then th ..."
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In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound ρ so that any frame set within ρ from F has the same property. In particular this proves a recent construction for the case m = 4n − 4 s stable under perturbations. Additionally we provide estimates of the stability radius.
(Chair)
, 2014
"... or the United States Government. This material is declared a work of the U.S. ..."
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or the United States Government. This material is declared a work of the U.S.
A Geometric Analysis of Phase Retrieval
"... Abstract Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, y k = a * k x for k = 1, . . . , m, is it possible to recover x ∈ C n (i.e., lengthn complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in vario ..."
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Abstract Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, y k = a * k x for k = 1, . . . , m, is it possible to recover x ∈ C n (i.e., lengthn complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines, and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretical explanations. In this paper, we take a step towards bridging this gap. We prove that when the measurement vectors a k 's are generic (i.i.d. complex Gaussian) and the number of measurements is large enough (m ≥ Cn log 3 n), with high probability, a natural leastsquares formulation for GPR has the following benign geometric structure: (1) there are no spurious local minimizers, and all global minimizers are equal to the target signal x, up to a global phase; and (2) the objective function has a negative curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a secondorder trustregion algorithm.
Simultaneously Structured Models with Application to Sparse and Lowrank Matrices
, 2012
"... The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal p ..."
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The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and lowrank. An important application is the sparse phase retrieval problem, where the goal is to recover a sparse signal from phaseless measurements. In machine learning, the problem comes up when combining several regularizers that each promote a certain desired structure. Often penalties (norms) that promote each individual structure are known and yield an orderwise optimal number of measurements (e.g., ℓ1 norm for sparsity, nuclear norm for matrix rank), soit isreasonabletominimize acombinationofsuchnorms. We showthat, surprisingly,if we use multiobjective optimization with the individual norms, then we can do no better, orderwise, than an algorithm that exploits only one of the several structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e., not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and lowrank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the ℓ1 and nuclear norms requires many more measurements. This proves an orderwise gap between the performance of the convex and nonconvex recovery problems in this case.
Contemporary Mathematics On Lipschitz Inversion of Nonlinear Redundant Representations
"... Abstract. In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α: H → Rm is injective, with (α(x))k = 〈x, fk〉2, where {f ..."
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Abstract. In this note we show that reconstruction from magnitudes of frame coefficients (the so called “phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α: H → Rm is injective, with (α(x))k = 〈x, fk〉2, where {f1, · · · , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω: Rm → H that is Lipschitz continuous. Additionally we obtain that the Lipschitz constant of this inverse map is at most 12 divided by the lower Lipschitz constant of α. 1.
Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
, 2014
"... Many maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. Semidefinite relaxations are among the most popular. Sometimes, the relaxations turn out to be tig ..."
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Many maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. Semidefinite relaxations are among the most popular. Sometimes, the relaxations turn out to be tight. In this paper, we study such a phenomenon. The angular synchronization problem consists in estimating a collection of n phases, given noisy measurements of some of the pairwise relative phases. The MLE for the angular synchronization problem is the solution of a (hard) nonbipartite Grothendieck problem over the complex numbers. It is known that its semidefinite relaxation enjoys worstcase approximation guarantees. In this paper, we consider a stochastic model on the input of that semidefinite relaxation. We assume there is a planted signal (corresponding to a ground truth set of phases) and the measurements are corrupted with random noise. Even though the MLE does not coincide with the planted signal, we show that the relaxation is, with high probability, tight. This holds even for high levels of noise. This analysis explains, for the interesting case of angular synchronization, a phenomenon which has been observed without explanation in many other settings. Namely, the fact that even when exact recovery of the ground truth is impossible, semidefinite relaxations for the MLE tend to be tight (in favorable noise regimes).