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Simultaneously Structured Models with Application to Sparse and Low-rank Matrices
, 2014
"... The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank 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 well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank 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 low-rank. Often norms that promote each individual structure are known, and allow for recovery using an order-wise 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 multi-objective optimization with these norms, then we can do no better, order-wise, 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 low-rank 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 order-wise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structure-inducing 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 low-rank tensor completion.
Sparsity constrained nonlinear optimization: Optimality conditions and algorithms, arXiv preprint arXiv:1203.4580
, 2012
"... This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinate-wise optimality. These conditions are then used to de ..."
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Cited by 33 (9 self)
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This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinate-wise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparse-simplex methods. The first algorithm is essentially a gradient projection method while the remaining two algorithms are of coordinate descent type. The theoretical convergence of these methods and their relations to the derived optimality conditions are studied. The algorithms and results are illustrated by several numerical examples. 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 18 (6 self)
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The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its
CPRL – An Extension of Compressive Sensing to the Phase Retrieval Problem
"... While compressive sensing (CS) has been one of the most vibrant research fields in the past few years, most development only applies to linear models. This limits its application in many areas where CS could make a difference. This paper presents a novel extension of CS to the phase retrieval proble ..."
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Cited by 8 (2 self)
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While compressive sensing (CS) has been one of the most vibrant research fields in the past few years, most development only applies to linear models. This limits its application in many areas where CS could make a difference. This paper presents a novel extension of CS to the phase retrieval problem, where intensity measurements of a linear system are used to recover a complex sparse signal. We propose a novel solution using a lifting technique – CPRL, which relaxes the NP-hard problem to a nonsmooth semidefinite program. Our analysis shows that CPRL inherits many desirable properties from CS, such as guarantees for exact recovery. We further provide scalable numerical solvers to accelerate its implementation. 1
Vectorial Phase Retrieval of 1-D Signals
"... Abstract—Reconstruction of signals from measurements of their spectral intensities, also known as the phase retrieval problem, is of fundamental importance in many scientific fields. In this paper we present a novel framework, denoted as vectorial phase retrieval, for reconstruction of pairs of sign ..."
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Cited by 5 (1 self)
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Abstract—Reconstruction of signals from measurements of their spectral intensities, also known as the phase retrieval problem, is of fundamental importance in many scientific fields. In this paper we present a novel framework, denoted as vectorial phase retrieval, for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference. We show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal. First, we prove that for compactly supported signals, in the absence of measurement noise, this new setup admits a unique solution. Next, we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it. Finally, we illustrate via simulations, that our algorithm can accurately reconstruct signals even at considerable noise levels. Index Terms—Convex relaxation, 1-D phase retrieval, signal recovery from modulus Fourier measurements, statistical model selection. I. INTRODUCTION AND MAIN RESULTS
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 multi-layered, involving multiple ran ..."
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Cited by 1 (1 self)
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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 multi-layered, 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 squaredmagnitude-only measurements, corrupted by additive noise or outliers.
Nonlinear Basis Pursuit
"... Abstract—In compressive sensing, the basis pursuit algorithm aims to find the sparsest solution to an underdetermined linear equation system. In this paper, we generalize basis pursuit to finding the sparsest solution to higher order nonlinear systems of equations, called nonlinear basis pursuit. In ..."
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Cited by 1 (0 self)
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Abstract—In compressive sensing, the basis pursuit algorithm aims to find the sparsest solution to an underdetermined linear equation system. In this paper, we generalize basis pursuit to finding the sparsest solution to higher order nonlinear systems of equations, called nonlinear basis pursuit. In contrast to the ex-isting nonlinear compressive sensing methods, the new algorithm that solves the nonlinear basis pursuit problem is convex and not greedy. The novel algorithm enables the compressive sensing approach to be used for a broader range of applications where there are nonlinear relationships between the measurements and the unknowns. I.
AN ALGORITHM FOR EXACT SUPER-RESOLUTION AND PHASE RETRIEVAL
"... We explore a fundamental problem of super-resolving a signal of interest from a few measurements of its low-pass magnitudes. We propose a 2-stage tractable algorithm that, in the absence of noise, admits perfect super-resolution of an r-sparse signal from 2r2 − 2r + 2 low-pass magnitude measurements ..."
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Cited by 1 (1 self)
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We explore a fundamental problem of super-resolving a signal of interest from a few measurements of its low-pass magnitudes. We propose a 2-stage tractable algorithm that, in the absence of noise, admits perfect super-resolution of an r-sparse signal from 2r2 − 2r + 2 low-pass magnitude measurements. The spike locations of the signal can assume any value over a continuous disk, without increasing the required sample size. The proposed algorithm first employs a conventional super-resolution algorithm (e.g. the matrix pencil approach) to recover unlabeled sets of signal correlation co-efficients, and then applies a simple sorting algorithm to disentangle and retrieve the true parameters in a deterministic manner. Our ap-proach can be adapted to multi-dimensional spike models and ran-dom Fourier sampling by replacing its first step with other harmonic retrieval algorithms.
Nonlinear compressive particle filtering
- in IEEE Conf. Decision and Control (CDC
, 2013
"... Abstract — Many systems for which compressive sensing is used today are dynamical. The common approach is to neglect the dynamics and see the problem as a sequence of independent problems. This approach has two disadvantages. Firstly, the temporal dependency in the state could be used to improve the ..."
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Cited by 1 (0 self)
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Abstract — Many systems for which compressive sensing is used today are dynamical. The common approach is to neglect the dynamics and see the problem as a sequence of independent problems. This approach has two disadvantages. Firstly, the temporal dependency in the state could be used to improve the accuracy of the state estimates. Secondly, having an estimate for the state and its support could be used to reduce the computational load of the subsequent step. In the linear Gaussian setting, compressive sensing was recently combined with the Kalman filter to mitigate above disadvantages. In the nonlinear dynamical case, compressive sensing can not be used and, if the state dimension is high, the particle filter would perform poorly. In this paper we combine one of the most novel developments in compressive sensing, nonlinear compressive sensing, with the particle filter. We show that the marriage of the two is essential and that neither the particle filter or nonlinear compressive sensing alone gives a satisfying solution. I.
