Results 1  10
of
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 ..."
Abstract

Cited by 28 (8 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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
Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study
, 2015
"... We study the Kaczmarz methods for solving systems of phaseless equations, i.e., the generalized phase retrieval problem. The methods extend the Kaczmarz methods for solving systems of linear equations by integrating a phase selection heuristic in each iteration and overall have the same per iterati ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We study the Kaczmarz methods for solving systems of phaseless equations, i.e., the generalized phase retrieval problem. The methods extend the Kaczmarz methods for solving systems of linear equations by integrating a phase selection heuristic in each iteration and overall have the same per iteration computational complexity. Extensive empirical performance comparisons establish the computational advantages of the Kaczmarz methods over other stateoftheart phase retrieval algorithms both in terms of the number of measurements needed for successful recovery and in terms of computation time. Preliminary convergence analysis is presented for the randomized Kaczmarz methods.
Algorithms and theory for clustering . . .
, 2014
"... In this dissertation we discuss three problems characterized by hidden structure or information. The first part of this thesis focuses on extracting subspace structures from data. Subspace Clustering is the problem of finding a multisubspace representation that best fits a collection of points tak ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
In this dissertation we discuss three problems characterized by hidden structure or information. The first part of this thesis focuses on extracting subspace structures from data. Subspace Clustering is the problem of finding a multisubspace representation that best fits a collection of points taken from a highdimensional space. As with most clustering problems, popular techniques for subspace clustering are often difficult to analyze theoretically as they are often nonconvex in nature. Theoretical analysis of these algorithms becomes even more challenging in the presence of noise and missing data. We introduce a collection of subspace clustering algorithms, which are tractable and provably robust to various forms of data imperfections. We further illustrate our methods with numerical experiments on a wide variety of data segmentation problems. In the second part of the thesis, we consider the problem of recovering the seemingly hidden phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We formulate the