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368
KSVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation
, 2006
"... In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signalatoms, signals are described by sparse linear combinations of these atoms. Applications that use sparse representation are many and inc ..."
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Cited by 935 (41 self)
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In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signalatoms, signals are described by sparse linear combinations of these atoms. Applications that use sparse representation are many and include compression, regularization in inverse problems, feature extraction, and more. Recent activity in this field has concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a given dictionary. Designing dictionaries to better fit the above model can be done by either selecting one from a prespecified set of linear transforms or adapting the dictionary to a set of training signals. Both of these techniques have been considered, but this topic is largely still open. In this paper we propose a novel algorithm for adapting dictionaries in order to achieve sparse signal representations. Given a set of training signals, we seek the dictionary that leads to the best representation for each member in this set, under strict sparsity constraints. We present a new method—the KSVD algorithm—generalizing the umeans clustering process. KSVD is an iterative method that alternates between sparse coding of the examples based on the current dictionary and a process of updating the dictionary atoms to better fit the data. The update of the dictionary columns is combined with an update of the sparse representations, thereby accelerating convergence. The KSVD algorithm is flexible and can work with any pursuit method (e.g., basis pursuit, FOCUSS, or matching pursuit). We analyze this algorithm and demonstrate its results both on synthetic tests and in applications on real image data.
Stable recovery of sparse overcomplete representations in the presence of noise
 IEEE TRANS. INFORM. THEORY
, 2006
"... Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes t ..."
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Cited by 460 (22 self)
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Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimalsparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combin ..."
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Cited by 427 (36 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Sparse Reconstruction by Separable Approximation
, 2007
"... Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution and reconstruction, and compressed sensing ..."
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Cited by 373 (38 self)
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Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution and reconstruction, and compressed sensing (CS) are a few wellknown areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (ℓ2) error term added to a sparsityinducing (usually ℓ1) regularizer. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex, sparsityinducing function. We propose iterative methods in which each step is an optimization subproblem involving a separable quadratic term (diagonal Hessian) plus the original sparsityinducing term. Our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. In addition to solving the standard ℓ2 − ℓ1 case, our approach handles other problems, e.g., ℓp regularizers with p � = 1, or groupseparable (GS) regularizers. Experiments with CS problems show that our approach provides stateoftheart speed for the standard ℓ2 − ℓ1 problem, and is also efficient on problems with GS regularizers. Index Terms — sparse approximation, compressed sensing, optimization, reconstruction.
Sparse solutions to linear inverse problems with multiple measurement vectors
 IEEE Trans. Signal Processing
, 2005
"... Abstract—We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known ..."
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Cited by 272 (22 self)
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Abstract—We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known to be NPhard, many single–measurement suboptimal algorithms have been formulated that have found utility in many different applications. Here, we consider in depth the extension of two classes of algorithms–Matching Pursuit (MP) and FOCal Underdetermined System Solver (FOCUSS)–to the multiple measurement case so that they may be used in applications such as neuromagnetic imaging, where multiple measurement vectors are available, and solutions with a common sparsity structure must be computed. Cost functions appropriate to the multiple measurement problem are developed, and algorithms are derived based on their minimization. A simulation study is conducted on a testcase dictionary to show how the utilization of more than one measurement vector improves the performance of the MP and FOCUSS classes of algorithm, and their performances are compared. I.
Iteratively reweighted algorithms for compressive sensing
 in 33rd International Conference on Acoustics, Speech, and Signal Processing (ICASSP
, 2008
"... The theory of compressive sensing has shown that sparse signals can be reconstructed exactly from many fewer measurements than traditionally believed necessary. In [1], it was shown empirically that using ℓ p minimization with p < 1 can do so with fewer measurements than with p = 1. In this paper ..."
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Cited by 185 (8 self)
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The theory of compressive sensing has shown that sparse signals can be reconstructed exactly from many fewer measurements than traditionally believed necessary. In [1], it was shown empirically that using ℓ p minimization with p < 1 can do so with fewer measurements than with p = 1. In this paper we consider the use of iteratively reweighted algorithms for computing local minima of the nonconvex problem. In particular, a particular regularization strategy is found to greatly improve the ability of a reweighted leastsquares algorithm to recover sparse signals, with exact recovery being observed for signals that are much less sparse than required by an unregularized version (such as FOCUSS, [2]). Improvements are also observed for the reweightedℓ 1 approach of [3]. Index Terms — Compressive sensing, signal reconstruction, nonconvex optimization, iteratively reweighted least squares, ℓ 1 minimization. 1.
Sparse Bayesian learning for basis selection
 IEEE Transactions on Signal Processing
, 2004
"... Abstract—Sparse Bayesian learning (SBL) and specifically relevance vector machines have received much attention in the machine learning literature as a means of achieving parsimonious representations in the context of regression and classification. The methodology relies on a parameterized prior tha ..."
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Cited by 150 (10 self)
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Abstract—Sparse Bayesian learning (SBL) and specifically relevance vector machines have received much attention in the machine learning literature as a means of achieving parsimonious representations in the context of regression and classification. The methodology relies on a parameterized prior that encourages models with few nonzero weights. In this paper, we adapt SBL to the signal processing problem of basis selection from overcomplete dictionaries, proving several results about the SBL cost function that elucidate its general behavior and provide solid theoretical justification for this application. Specifically, we have shown that SBL retains a desirable property of the 0norm diversity measure (i.e., the global minimum is achieved at the maximally sparse solution) while often possessing a more limited constellation of local minima. We have also demonstrated that the local minima that do exist are achieved at sparse solutions. Later, we provide a novel interpretation of SBL that gives us valuable insight into why it is successful in producing sparse representations. Finally, we include simulation studies comparing sparse Bayesian learning with Basis Pursuit and the more recent FOCal Underdetermined System Solver (FOCUSS) class of basis selection algorithms. These results indicate that our theoretical insights translate directly into improved performance. Index Terms—Basis selection, diversity measures, linear inverse problems, sparse Bayesian learning, sparse representations. I.
Enhancing Sparsity by Reweighted ℓ1 Minimization
, 2007
"... It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many si ..."
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Cited by 145 (4 self)
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It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ1minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed nearsparsity in overcomplete representations—not by reweighting the ℓ1 norm of the coefficient sequence as is common, but by reweighting the ℓ1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as compressed sensing.
Bayesian compressive sensing via belief propagation
 IEEE Trans. Signal Processing
, 2010
"... Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can comple ..."
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Cited by 125 (19 self)
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Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform approximate Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast encoding and decoding is provided using sparse encoding matrices, which also improve BP convergence by reducing the presence of loops in the graph. To decode a lengthN signal containing K large coefficients, our CSBP decoding algorithm uses O(K log(N)) measurements and O(N log 2 (N)) computation. Finally, sparse encoding matrices and the CSBP decoding algorithm can be modified to support a variety of signal models and measurement noise. 1
An affine scaling methodology for best basis selection
 IEEE TRANS. SIGNAL PROCESSING
, 1999
"... A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser and Donoho. These measures include the pnormlike (`(p 1)) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a f ..."
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Cited by 124 (21 self)
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A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser and Donoho. These measures include the pnormlike (`(p 1)) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the Affine Scaling Transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the `(p 1) diversity measures are equivalent to a recently developed class of algorithms called FOCal Underdetermined System Solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a wellbehaved p =0normlike optimization algorithm. Computer experiments demonstrate that the pnormlike and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution.