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19
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.
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
Phase retrieval of sparse signals using optimization transfer and ADMM
 in Proc. IEEE Intl. Conf. on Image Processing, 2014
"... ABSTRACT We propose a reconstruction method for the phase retrieval problem prevalent in optics, crystallography, and other imaging applications. Our approach uses signal sparsity to provide robust reconstruction, even in the presence of outliers. Our method is multilayered, involving multiple ran ..."
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ABSTRACT We propose a reconstruction method for the phase retrieval problem prevalent in optics, crystallography, and other imaging applications. Our approach uses signal sparsity to provide robust reconstruction, even in the presence of outliers. Our method is multilayered, involving multiple random initial conditions, convex majorization, variable splitting, and alternating directions method of multipliers (ADMM)based implementation. Monte Carlo simulations demonstrate that our algorithm can correctly and robustly detect sparse signals from full and undersampled sets of squaredmagnitudeonly measurements, corrupted by additive noise or outliers.
Sparse Phase Retrieval from ShortTime Fourier Measurements
"... Abstract—We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the shorttime Fourier transform (STFT). We first show that the redundancy offere ..."
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Abstract—We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the shorttime Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements. Index Terms—GESPAR, phase retrieval, shorttime Fourier transform, sparsity.
Proximal heterogeneous block inputoutput method and application to blind ptychographic diffraction imaging, arXiv:1408.1887v1
"... We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literatu ..."
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We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practically verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current stateoftheart on simulated and experimental data validates our approach and points toward directions for further improvement.
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 ..."
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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
Undersampled Phase Retrieval with Outliers
 SUBMITTED TO IEEE TRANSACTIONS ON COMPUTATIONAL IMAGING
, 2014
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Scalable Convex Methods for Phase Retrieval
"... Abstract—This paper describes scalable convex optimization methods for phase retrieval. The main characteristics of these methods are the cheap periteration complexity and the lowmemory footprint. With a variant of the original PhaseLift formulation, we first illustrate how to leverage the scalabl ..."
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Abstract—This paper describes scalable convex optimization methods for phase retrieval. The main characteristics of these methods are the cheap periteration complexity and the lowmemory footprint. With a variant of the original PhaseLift formulation, we first illustrate how to leverage the scalable FrankWolfe (FW) method (also known as the conditional gradient algorithm), which requires a tuning parameter. We demonstrate that we can estimate the tuning parameter of the FW algorithm directly from the measurements, with rigorous theoretical guarantees. We then illustrate numerically that recent advances in universal primaldual convex optimization methods offer significant scalability improvements over the FW method, by recovering full HD resolution color images from their quadratic measurements. I.
scattered light intensity
"... Deciphering the threedimensional (3D) structure of complex molecules is of major importance, typically accomplished with Xray crystallography. Unfortunately, many important molecules cannot be crystallized, hence their 3D structure is unknown. Ankylography presents an alternative, relying on scatt ..."
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Deciphering the threedimensional (3D) structure of complex molecules is of major importance, typically accomplished with Xray crystallography. Unfortunately, many important molecules cannot be crystallized, hence their 3D structure is unknown. Ankylography presents an alternative, relying on scattering an ultrashort Xray pulse off a single molecule before it disintegrates, measuring the farfield intensity on a twodimensional surface, followed by computation. However, significant information is absent due to lower dimensionality of the measurements and the inability to measure the phase. Recent Ankylography experiments attracted much interest, but it was counterargued that Ankylography is valid only for objects containing a small number of volume pixels. Here, we propose a sparsitybased approach to reconstruct the 3D structure of molecules. Sparsity is natural for Ankylography, because molecules can be represented compactly in stoichiometric basis. Utilizing sparsity, we surpass current limits on recoverable information by orders of magnitude, paving the way for deciphering the 3D structure of macromolecules.