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21
Compressive Phase Retrieval via Generalized Approximate Message Passing
"... Abstract—In this paper, we propose a novel approach to compressive phase retrieval based on loopy belief propagation and, in particular, on the generalized approximate message passing (GAMP) algorithm. Numerical results show that the proposed PRGAMP algorithm has excellent phasetransition behavior ..."
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Abstract—In this paper, we propose a novel approach to compressive phase retrieval based on loopy belief propagation and, in particular, on the generalized approximate message passing (GAMP) algorithm. Numerical results show that the proposed PRGAMP algorithm has excellent phasetransition behavior, noise robustness, and runtime. In particular, for successful recovery of synthetic BernoullicircularGaussian signals, PRGAMP requires ≈ 4 times the number of measurements as a phaseoracle version of GAMP and, at moderate to large SNR, the NMSE of PRGAMP is only ≈ 3 dB worse than that of phaseoracle GAMP. A comparison to the recently proposed convexrelation approach known as “CPRL ” reveals PRGAMP’s superior phase transition and ordersofmagnitude faster runtimes, especially as the problem dimensions increase. When applied to the recovery of a 65kpixel grayscale image from 32k randomly masked magnitude measurements, numerical results show a median PRGAMP runtime of only 13.4 seconds. A. Phase retrieval I.
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 24 (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.
Lowrank Solutions of Linear Matrix Equations via Procrustes Flow
, 2015
"... In this paper we study the problem of recovering an lowrank positive semidefinite matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a nonconvex objective. We show that as ..."
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Cited by 4 (0 self)
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In this paper we study the problem of recovering an lowrank positive semidefinite matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a nonconvex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a n×n matrix of rank r when the number of measurements exceeds a constant times nr. 1
Quantization and greed are good: One bit phase retrieval, robustness and greedy refinements. arXiv preprint arXiv:1312.1830
, 2013
"... In this paper, we study the problem of robust phase recovery. We investigate a novel approach based on extremely quantized (onebit) measurements and a corresponding recovery scheme. The proposed approach has surprising robustness properties and, unlike currently available methods, allows to effici ..."
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Cited by 4 (1 self)
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In this paper, we study the problem of robust phase recovery. We investigate a novel approach based on extremely quantized (onebit) measurements and a corresponding recovery scheme. The proposed approach has surprising robustness properties and, unlike currently available methods, allows to efficiently perform phase recovery from measurements affected by severe (possibly unknown) non linear perturbations, such as distortions (e.g. clipping). Beyond robustness, we show how our approach can be used within greedy approaches based on alternating minimization. In particular, we propose novel initialization schemes for the alternating minimization achieving favorable convergence properties with improved sample complexity. 1
Sharp timedata tradeoffs for linear inverse problems
 In preparation
, 2015
"... In this paper we characterize sharp timedata tradeoffs for optimization problems used for solving linear inverse problems. We focus on the minimization of a leastsquares objective subject to a constraint defined as the sublevel set of a penalty function. We present a unified convergence analysis ..."
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Cited by 3 (2 self)
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In this paper we characterize sharp timedata tradeoffs for optimization problems used for solving linear inverse problems. We focus on the minimization of a leastsquares objective subject to a constraint defined as the sublevel set of a penalty function. We present a unified convergence analysis of the gradient projection algorithm applied to such problems. We sharply characterize the convergence rate associated with a wide variety of random measurement ensembles in terms of the number of measurements and structural complexity of the signal with respect to the chosen penalty function. The results apply to both convex and nonconvex constraints, demonstrating that a linear convergence rate is attainable even though the least squares objective is not strongly convex in these settings. When specialized to Gaussian measurements our results show that such linear convergence occurs when the number of measurements is merely 4 times the minimal number required to recover the desired signal at all (a.k.a. the phase transition). We also achieve a slower but geometric rate of convergence precisely above the phase transition point. Extensive numerical results suggest that the derived rates exactly match the empirical performance.
Phasecode: Fast and efficient compressive phase retrieval based on sparsegraphcodes,” arXiv preprint arXiv:1408.0034
, 2014
"... We consider the problem of recovering a complex signal x ∈ Cn from m intensity measurements of the form aix, 1 ≤ i ≤ m, where ai is a measurement row vector. We address multiple settings corresponding to whether the measurement vectors are unconstrained choices or not, and to whether the signal to ..."
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Cited by 2 (0 self)
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We consider the problem of recovering a complex signal x ∈ Cn from m intensity measurements of the form aix, 1 ≤ i ≤ m, where ai is a measurement row vector. We address multiple settings corresponding to whether the measurement vectors are unconstrained choices or not, and to whether the signal to be recovered is sparse or not. However, our main focus is on the case where the measurement vectors are unconstrained, and where x is exactly Ksparse, or the socalled general compressive phaseretrieval problem. We introduce PhaseCode, a novel family of fast and efficient mergeandcolor algorithms (that includes Unicolor PhaseCode and Multicolor PhaseCode) that are based on a sparsegraphcodes framework. As one instance, our Unicolor PhaseCode algorithm can provably recover, with high probability, all but a tiny 10−7 fraction of the significant signal components, using at most m = 14K measurements, which is a small constant factor from the fundamental limit, with an optimal O(K) decoding time and an optimal O(K) memory complexity. Next, motivated by some important practical classes of optical systems, we consider a “Fourierfriendly ” constrained measurement setting, and show that its performance matches that of the unconstrained setting. In the Fourierfriendly setting that we consider, the
Robust phase retrieval and superresolution from one bit coded diffraction patterns. arXiv preprint arXiv:1402.2255
, 2014
"... In this paper we study a realistic setup for phase retrieval, where the signal of interest is modulated or masked and then for each modulation or mask a diffraction pattern is collected, producing a coded diffraction pattern (CDP) [CLM13]. We are interested in the setup where the resolution of the c ..."
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In this paper we study a realistic setup for phase retrieval, where the signal of interest is modulated or masked and then for each modulation or mask a diffraction pattern is collected, producing a coded diffraction pattern (CDP) [CLM13]. We are interested in the setup where the resolution of the collected CDP is limited by the Fraunhofer diffraction limit of the imaging system. We investigate a novel approach based on a geometric quantization scheme of phaseless linear measurements into (onebit) coded diffraction patterns, and a corresponding recovery scheme. The key novelty in this approach consists in comparing pairs of coded diffractions patterns across frequencies: the one bit measurements obtained rely on the order statistics of the unquantized measurements rather than their values. This results in a robust phase recovery, and unlike currently available methods, allows to efficiently perform phase recovery from measurements affected by severe (possibly unknown) non linear, rank preserving perturbations, such as distortions. Another important feature of this approach consists in the fact that it enables also superresolution and blinddeconvolution, beyond the diffraction limit of a given imaging system. 1
Solving Random Quadratic Systems of Equations is nearly as easy as . . .
, 2015
"... We consider the fundamental problem of solving quadratic systems of equations in n variables, where yi = 〈ai,x〉2, i = 1,...,m and x ∈ Rn is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional ..."
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Cited by 2 (1 self)
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We consider the fundamental problem of solving quadratic systems of equations in n variables, where yi = 〈ai,x〉2, i = 1,...,m and x ∈ Rn is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach [11]. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data {ai} and {yi} as soon as the ratio m/n between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have yi ≈ 〈ai,x〉2 and prove that our algorithms achieve a statistical accuracy, which is nearly unimprovable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size—hence the title of this paper. For instance, we