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69
Simultaneously Structured Models with Application to Sparse and Lowrank Matrices
, 2014
"... 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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Cited by 41 (5 self)
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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. Often norms that promote each individual structure are known, and allow for recovery using an orderwise optimal number of measurements (e.g., `1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multiobjective optimization with these norms, then we can do no better, orderwise, than an algorithm that exploits only one of the present 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, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the `1 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. Our framework applies to arbitrary structureinducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and lowrank tensor completion.
Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming
, 2011
"... Abstract: Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, ℓ1minimization, guarantees recovery ..."
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Cited by 37 (5 self)
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Abstract: Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely, ℓ1minimization, guarantees recovery of sparse parameter signals even when the system is underdetermined. In this paper, we consider a more challenging problem: when the phase of the output measurements from a linear system is omitted. Using a lifting technique, we show that even though the phase information is missing, the sparse signal can be recovered exactly by solving a semidefinite program when the sampling rate is sufficiently high. This is an interesting finding since the exact solutions to both sparse signal recovery and phase retrieval are combinatorial. The results extend the type of applications that compressive sensing can be applied to those where only output magnitudes can be observed. We demonstrate the accuracy of the algorithms through extensive simulation and a practical experiment.
Phase Retrieval from Coded Diffraction Patterns
, 2013
"... This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the inten ..."
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Cited by 21 (5 self)
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This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the intensity of its diffraction pattern, each modulation thereby producing a sort of coded diffraction pattern. We show that PhaseLift, a recent convex programming technique, recovers the phase information exactly from a number of random modulations, which is polylogarithmic in the number of unknowns. Numerical experiments with noiseless and noisy data complement our theoretical analysis and illustrate our approach.
Phase retrieval with polarization
 SIAM J. ON IMAGING SCI
, 2013
"... In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and ..."
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Cited by 20 (4 self)
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In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [14]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternatingprojectionstype procedure.
Phase Retrieval via Wirtinger Flow: Theory and Algorithms
, 2014
"... We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complexvalued signal x ∈ Cn about which we have phaseless samples of the form yr = ∣⟨ar,x⟩∣2, r = 1,...,m (knowledge of the phase of these samples would yield a linear system). This pape ..."
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Cited by 18 (4 self)
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We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complexvalued signal x ∈ Cn about which we have phaseless samples of the form yr = ∣⟨ar,x⟩∣2, r = 1,...,m (knowledge of the phase of these samples would yield a linear system). This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a nearlinear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval.
Phase retrieval using alternating minimization
 In NIPS
, 2013
"... Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. Over the last two decades, a popular generic empirical approach to the many variants of this problem has been one of alternating minimization; i.e. alternating between estima ..."
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Cited by 17 (1 self)
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Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. Over the last two decades, a popular generic empirical approach to the many variants of this problem has been one of alternating minimization; i.e. alternating between estimating the missing phase information, and the candidate solution. In this paper, we show that a simple alternating minimization algorithm geometrically converges to the solution of one such problem – finding a vector x from y,A, where y = ATx  and z  denotes a vector of elementwise magnitudes of z – under the assumption that A is Gaussian. Empirically, our algorithm performs similar to recently proposed convex techniques for this variant (which are based on “lifting ” to a convex matrix problem) in sample complexity and robustness to noise. However, our algorithm is much more efficient and can scale to large problems. Analytically, we show geometric convergence to the solution, and sample complexity that is off by log factors from obvious lower bounds. We also establish close to optimal scaling for the case when the unknown vector is sparse. Our work represents the only known theoretical guarantee for alternating minimization for any variant of phase retrieval problems in the nonconvex setting. 1
Phase Retrieval with Application to Optical Imaging
, 2015
"... The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its ..."
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Cited by 16 (5 self)
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The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its
Determination of all pure quantum states from a minimal number of observables
, 2014
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A Partial Derandomization of PhaseLift using Spherical Designs
, 2013
"... ABSTRACT. The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally efficient, numerically stable, and comes with rigorous pe ..."
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Cited by 11 (4 self)
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ABSTRACT. The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally efficient, numerically stable, and comes with rigorous performance guarantees. PhaseLift is optimal in the sense that the number of amplitude measurements required for phase reconstruction scales linearly with the dimension of the signal. However, it specifically demands Gaussian random measurement vectors — a limitation that restricts practical utility and obscures the specific properties of measurement ensembles that enable phase retrieval. Here we present a partial derandomization of PhaseLift that only requires sampling from certain polynomial size vector configurations, called tdesigns. Such configurations have been studied in algebraic combinatorics, coding theory, and quantum information. We prove reconstruction guarantees for a number of measurements that depends on the degree t of the design. If the degree is allowed to to grow logarithmically with the dimension, the bounds become tight up to polylogfactors. Beyond the specific case of PhaseLift, this work highlights the utility of spherical designs for the derandomization of data recovery schemes. 1.
Convex recovery from interferometric measurements. arXiv preprint arXiv:1307.6864
, 2013
"... This note formulates a deterministic recovery result for vectors x from quadratic measurements of the form (Ax)i(Ax)j for some leftinvertible A. Recovery is exact, or stable in the noisy case, when the couples (i, j) are chosen as edges of a wellconnected graph. One possible way of obtaining the ..."
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Cited by 9 (0 self)
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This note formulates a deterministic recovery result for vectors x from quadratic measurements of the form (Ax)i(Ax)j for some leftinvertible A. Recovery is exact, or stable in the noisy case, when the couples (i, j) are chosen as edges of a wellconnected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a dataweighted graph Laplacian. Such quadratic measurements have found applications in phase retrieval, angular synchronization, and more recently interferometric waveform inversion. Acknowledgments. The authors would like to thank Amit Singer for interesting discussions. 1