Results 1  10
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108
Distributed compressed sensing
, 2005
"... Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algori ..."
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Cited by 135 (26 self)
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Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algorithms for multisignal ensembles that exploit both intra and intersignal correlation structures. The DCS theory rests on a new concept that we term the joint sparsity of a signal ensemble. We study in detail three simple models for jointly sparse signals, propose algorithms for joint recovery of multiple signals from incoherent projections, and characterize theoretically and empirically the number of measurements per sensor required for accurate reconstruction. We establish a parallel with the SlepianWolf theorem from information theory and establish upper and lower bounds on the measurement rates required for encoding jointly sparse signals. In two of our three models, the results are asymptotically bestpossible, meaning that both the upper and lower bounds match the performance of our practical algorithms. Moreover, simulations indicate that the asymptotics take effect with just a moderate number of signals. In some sense DCS is a framework for distributed compression of sources with memory, which has remained a challenging problem for some time. DCS is immediately applicable to a range of problems in sensor networks and arrays.
Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation
"... In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relax ..."
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Cited by 100 (22 self)
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In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worstcase analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed ℓ2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.
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 97 (14 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.
Compressive Sensing Principles and Iterative Sparse Recovery for Inverse and IllPosed Problems
, 2010
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Bregman iterative algorithms for ℓ1minimization with applications to compressed sensing
 SIAM J. IMAGING SCI
, 2008
"... We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of 1 insta ..."
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Cited by 83 (15 self)
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We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of 1 instances of the unconstrained problem minu∈Rn μ‖u‖1 + 2 ‖Au−fk ‖ 2 2 for given matrix A and vector f k. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrixvector operations involving A and A ⊤ can be computed by fast transforms. Utilizing a fast fixedpoint continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.
Visual classification with multitask joint sparse representation
 In CVPR
, 2010
"... Abstract — We address the problem of visual classification with multiple features and/or multiple instances. Motivated by the recent success of multitask joint covariate selection, we formulate this problem as a multitask joint sparse representation model to combine the strength of multiple features ..."
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Cited by 64 (1 self)
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Abstract — We address the problem of visual classification with multiple features and/or multiple instances. Motivated by the recent success of multitask joint covariate selection, we formulate this problem as a multitask joint sparse representation model to combine the strength of multiple features and/or instances for recognition. A joint sparsityinducing norm is utilized to enforce classlevel joint sparsity patterns among the multiple representation vectors. The proposed model can be efficiently optimized by a proximal gradient method. Furthermore, we extend our method to the setup where features are described in kernel matrices. We then investigate into two applications of our method to visual classification: 1) fusing multiple kernel features for object categorization and 2) robust face recognition in video with an ensemble of query images. Extensive experiments on challenging realworld data sets demonstrate that the proposed method is competitive to the stateoftheart methods in respective applications. Index Terms — Feature fusion, multitask learning, sparse representation, visual classification.
Blockwise Coordinate Descent Procedures for the Multitask Lasso, with Applications to Neural Semantic Basis Discovery
"... We develop a cyclical blockwise coordinate descent algorithm for the multitask Lasso that efficiently solves problems with thousands of features and tasks. The main result shows that a closedform Winsorization operator can be obtained for the supnorm penalized least squares regression. This allow ..."
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Cited by 60 (3 self)
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We develop a cyclical blockwise coordinate descent algorithm for the multitask Lasso that efficiently solves problems with thousands of features and tasks. The main result shows that a closedform Winsorization operator can be obtained for the supnorm penalized least squares regression. This allows the algorithm to find solutions to very largescale problems far more efficiently than existing methods. This result complements the pioneering work of Friedman, et al. (2007) for the singletask Lasso. As a case study, we use the multitask Lasso as a variable selector to discover a semantic basis for predicting human neural activation. The learned solution outperforms the standard basis for this task on the majority of test participants, while requiring far fewer assumptions about cognitive neuroscience. We demonstrate how this learned basis can yield insights into how the brain represents the meanings of words. 1.
Iterative thresholding algorithms
 in Preprint, 2007. [Online]. Available : http ://www.dsp.ece.rice.edu/cs
"... This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an a ..."
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Cited by 53 (10 self)
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This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an algorithm recently studied by the authors. We prove stability of minimizers with respect to the parameters of the functional and its regularization properties by means of Γconvergence. All investigations are done in the general setting of vectorvalued (multichannel) data.
1 Dictionary Learning for Sparse Approximations with the Majorization Method
"... Abstract—In order to find sparse approximations of signals, an appropriate generative model for the signal class has to be known. If the model is unknown, it can be adapted using a set of training samples. This paper presents a novel method for dictionary learning and extends the learning problem by ..."
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Cited by 49 (10 self)
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Abstract—In order to find sparse approximations of signals, an appropriate generative model for the signal class has to be known. If the model is unknown, it can be adapted using a set of training samples. This paper presents a novel method for dictionary learning and extends the learning problem by introducing different constraints on the dictionary. The convergence of the proposed method to a fixed point is guaranteed, unless the accumulation points form a continuum. This holds for different sparsity measures. The majorization method is an optimization method that substitutes the original objective function with a surrogate function that is updated in each optimization step. This method has been used successfully in sparse approximation and statistical estimation (e.g. Expectation Maximization (EM)) problems. This paper shows that the majorization method can be used for the dictionary learning problem too. The proposed method is compared with other methods on both synthetic and real data and different constraints on the dictionary are compared. Simulations show the advantages of the proposed method over other currently available dictionary learning methods not only in terms of average performance but also in terms of computation time.
LowDimensional Models for Dimensionality Reduction and Signal Recovery: A Geometric Perspective
, 2009
"... We compare and contrast from a geometric perspective a number of lowdimensional signal models that support stable informationpreserving dimensionality reduction. We consider sparse and compressible signal models for deterministic and random signals, structured sparse and compressible signal model ..."
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Cited by 47 (12 self)
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We compare and contrast from a geometric perspective a number of lowdimensional signal models that support stable informationpreserving dimensionality reduction. We consider sparse and compressible signal models for deterministic and random signals, structured sparse and compressible signal models, point clouds, and manifold signal models. Each model has a particular geometrical structure that enables signal information in to be stably preserved via a simple linear and nonadaptive projection to a much lower dimensional space whose dimension either is independent of the ambient dimension at best or grows logarithmically with it at worst. As a bonus, we point out a common misconception related to probabilistic compressible signal models, that is, that the generalized Gaussian and Laplacian random models do not support stable linear dimensionality reduction.