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Structured compressed sensing: From theory to applications
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard ..."
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Cited by 104 (16 self)
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Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuoustime signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
Hyperspectral unmixing overview: Geometrical, statistical, and sparse regressionbased approaches
 IEEE J. SEL. TOPICS APPL. EARTH OBSERV. REMOTE SENS
, 2012
"... Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher sp ..."
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Cited by 103 (34 self)
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Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, illposed
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.
Blind compressed sensing
 IEEE TRANS. INF. THEORY
, 2011
"... The fundamental principle underlying compressed sensing is that a signal, which is sparse under some basis representation, can be recovered from a small number of linear measurements. However, prior knowledge of the sparsity basis is essential for the recovery process. This work introduces the conc ..."
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Cited by 15 (3 self)
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The fundamental principle underlying compressed sensing is that a signal, which is sparse under some basis representation, can be recovered from a small number of linear measurements. However, prior knowledge of the sparsity basis is essential for the recovery process. This work introduces the concept of blind compressed sensing, which avoids the need to know the sparsity basis in both the sampling and the recovery process. We suggest three possible constraints on the sparsity basis that can be added to the problem in order to guarantee a unique solution. For each constraint, we prove conditions for uniqueness, and suggest a simple method to retrieve the solution. We demonstrate through simulations that our methods can achieve results similar to those of standard compressed sensing, which rely on prior knowledge of the sparsity basis, as long as the signals are sparse enough. This offers a general sampling and reconstruction system that fits all sparse signals, regardless of the sparsity basis, under the conditions and constraints presented in this work.
A NONCONVEX ADMM ALGORITHM FOR GROUP SPARSITY WITH SPARSE GROUPS
"... We present an efficient algorithm for computing sparse representations whose nonzero coefficients can be divided into groups, few of which are nonzero. In addition to this group sparsity, we further impose that the nonzero groups themselves be sparse. We use a nonconvex optimization approach for thi ..."
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Cited by 12 (2 self)
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We present an efficient algorithm for computing sparse representations whose nonzero coefficients can be divided into groups, few of which are nonzero. In addition to this group sparsity, we further impose that the nonzero groups themselves be sparse. We use a nonconvex optimization approach for this purpose, and use an efficient ADMM algorithm to solve the nonconvex problem. The efficiency comes from using a novel shrinkage operator, one that minimizes nonconvex penalty functions for enforcing sparsity and group sparsity simultaneously. Our numerical experiments show that combining sparsity and group sparsity improves signal reconstruction accuracy compared with either property alone. We also find that using nonconvex optimization significantly improves results in comparison with convex optimization. Index Terms — Sparse representations, group sparsity, shrinkage, nonconvex optimization, alternating direction method of multipliers 1.
Collaborative Sparse Regression For Hyperspectral Unmixing
 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
, 2013
"... Sparse unmixing has been recently introduced in hyperspectral imaging as a framework to characterize mixed pixels. It assumes that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on th ..."
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Cited by 11 (4 self)
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Sparse unmixing has been recently introduced in hyperspectral imaging as a framework to characterize mixed pixels. It assumes that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In this paper, we present a refinement of the sparse unmixing methodology recently introduced which exploits the usual very low number of endmembers present in real images, out of a very large library. Specifically, we adopt the collaborative (also called “multitask” or “simultaneous”) sparse regression framework that improves the unmixing results by solving a joint sparse regression problem, where the sparsity is simultaneously imposed to all pixels in the data set. Our experimental results with both synthetic and real hyperspectral data sets show clearly the advantages obtained using the new joint sparse regression strategy, compared with the pixelwise independent approach.
Social sparsity! neighborhood systems enrich structured shrinkage operators
 IEEE Trans. Signal Processing
, 2013
"... Abstract—Sparse and structured signal expansions on dictionaries can be obtained through explicit modeling in the coefficient domain. The originality of the present article lies in the construction and the study of generalized shrinkage operators, whose goal is to identify structured significance ma ..."
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Cited by 8 (2 self)
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Abstract—Sparse and structured signal expansions on dictionaries can be obtained through explicit modeling in the coefficient domain. The originality of the present article lies in the construction and the study of generalized shrinkage operators, whose goal is to identify structured significance maps and give rise to structured thresholding. These generalize Group Lasso and the previously introduced Elitist Lasso by introducing more flexibility in the coefficient domain modeling, and lead to the notion of social sparsity. The proposed operators are studied theoretically and embedded in iterative thresholding algorithms. Moreover, a link between these operators and a convex functional is established. Numerical studies on both simulated and real signals confirm the benefits of such an approach.
Decentralized sparsityregularized rank minimization: Algorithms and applications
 IEEE Trans. Signal Process
, 2013
"... Abstract—Given a limited number of entries from the superposition of a lowrank matrix plus the product of a known compression matrix times a sparse matrix, recovery of the lowrank and sparse components is a fundamental task subsuming compressed sensing, matrix completion, and principal components ..."
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Cited by 7 (5 self)
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Abstract—Given a limited number of entries from the superposition of a lowrank matrix plus the product of a known compression matrix times a sparse matrix, recovery of the lowrank and sparse components is a fundamental task subsuming compressed sensing, matrix completion, and principal components pursuit. This paper develops algorithms for decentralized sparsityregularized rank minimization over networks, when the nuclear andnorm are used as surrogates to the rank and nonzero entry counts of the sought matrices, respectively. While nuclearnorm minimization has welldocumented merits when centralized processing is viable, nonseparability of the singularvalue sum challenges its decentralized minimization. To overcome this limitation, leveraging an alternative characterization of the nuclear norm yields a separable, yet nonconvex cost minimized via the alternatingdirection method of multipliers. Interestingly, if the decentralized (nonconvex) estimator converges, under certain conditions it provably attains the global optimum of its centralized counterpart. As a result, this paper bridges the performance gap between centralized and innetwork decentralized, sparsityregularized rankminimization. This, in turn, facilitates (stable) recovery of the low rank and sparse model matrices through reducedcomplexity pernode computations, and affordable message passing among singlehop neighbors. Several application domains are outlined to highlight the generality and impact of the proposed framework. These include unveiling traffic anomalies in backbone networks, and predicting networkwide path latencies. Simulations with synthetic and real network data confirm the convergence of the novel decentralized algorithm, and its centralized performance guarantees. Index Terms—Decentralized optimization, sparsity, nuclear norm, low rank, networks, Lasso, matrix completion. I.
The Landmark Selection Method for Multiple Output Prediction
"... Conditional modeling x ↦ → y is a central problem in machine learning. A substantial research effort is devoted to such modeling when x is high dimensional. We consider, instead, the case of a high dimensional y, where x is either low dimensional or high dimensional. Our approach is based on selecti ..."
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Cited by 6 (0 self)
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Conditional modeling x ↦ → y is a central problem in machine learning. A substantial research effort is devoted to such modeling when x is high dimensional. We consider, instead, the case of a high dimensional y, where x is either low dimensional or high dimensional. Our approach is based on selecting a small subset yL of the dimensions of y, and proceed by modeling (i) x ↦ → yL and (ii) yL ↦ → y. Composingthesetwomodels,weobtainaconditionalmodelx ↦ → y thatpossesses convenient statistical properties. Multilabel classification and multivariate regression experiments on several datasets show that this method outperforms the one vs. all approach as well as several sophisticated multiple output prediction methods. 1.
Learning Efficient Structured Sparse Models
"... We present a comprehensive framework for structured sparse coding and modeling extending the recent ideas of using learnable fast regressors to approximate exact sparse codes. For this purpose, we propose an efficient feed forward architecture derived from the iteration of the blockcoordinate algor ..."
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Cited by 6 (3 self)
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We present a comprehensive framework for structured sparse coding and modeling extending the recent ideas of using learnable fast regressors to approximate exact sparse codes. For this purpose, we propose an efficient feed forward architecture derived from the iteration of the blockcoordinate algorithm. This architecture approximates the exact structured sparse codes with a fraction of the complexity of the standard optimization methods. We also show that by using different training objective functions, the proposed learnable sparse encoders are not only restricted to be approximants of the exact sparse code for a pregiven dictionary, but can be rather used as fullfeatured sparse encoders or even modelers. A simple implementation shows several orders of magnitude speedup compared to the stateoftheart exact optimization algorithms at minimal performance degradation, making the proposed framework suitable for real time and largescale applications. 1.