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613
Sharp thresholds for highdimensional and noisy sparsity recovery using l1constrained quadratic programmming (Lasso)
, 2006
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An interiorpoint method for largescale l1regularized logistic regression
 Journal of Machine Learning Research
, 2007
"... Logistic regression with ℓ1 regularization has been proposed as a promising method for feature selection in classification problems. In this paper we describe an efficient interiorpoint method for solving largescale ℓ1regularized logistic regression problems. Small problems with up to a thousand ..."
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Cited by 153 (6 self)
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Logistic regression with ℓ1 regularization has been proposed as a promising method for feature selection in classification problems. In this paper we describe an efficient interiorpoint method for solving largescale ℓ1regularized logistic regression problems. Small problems with up to a thousand or so features and examples can be solved in seconds on a PC; medium sized problems, with tens of thousands of features and examples, can be solved in tens of seconds (assuming some sparsity in the data). A variation on the basic method, that uses a preconditioned conjugate gradient method to compute the search step, can solve very large problems, with a million features and examples (e.g., the 20 Newsgroups data set), in a few minutes, on a PC. Using warmstart techniques, a good approximation of the entire regularization path can be computed much more efficiently than by solving a family of problems independently.
Compressed sensing and best kterm approximation
 J. Amer. Math. Soc
, 2009
"... Compressed sensing is a new concept in signal processing where one seeks to minimize the number of measurements to be taken from signals while still retaining the information necessary to approximate them well. The ideas have their origins in certain abstract results from functional analysis and app ..."
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Cited by 151 (11 self)
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Compressed sensing is a new concept in signal processing where one seeks to minimize the number of measurements to be taken from signals while still retaining the information necessary to approximate them well. The ideas have their origins in certain abstract results from functional analysis and approximation theory by Kashin [23] but were recently brought into the forefront by the work of Candès, Romberg and Tao [7, 5, 6] and Donoho [9] who constructed concrete algorithms and showed their promise in application. There remain several fundamental questions on both the theoretical and practical side of compressed sensing. This paper is primarily concerned about one of these theoretical issues revolving around just how well compressed sensing can approximate a given signal from a given budget of fixed linear measurements, as compared to adaptive linear measurements. More precisely, we consider discrete signals x ∈ IR N, allocate n < N linear measurements of x, and we describe the range of k for which these measurements encode enough information to recover x in the sense of ℓp to the accuracy of best kterm approximation. We also consider the problem of having such accuracy only with high probability.
Sparsity and Incoherence in Compressive Sampling
, 2006
"... We consider the problem of reconstructing a sparse signal x 0 ∈ R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0, where U is an orthonormal matrix, we show that ℓ1 minimization recovers x 0 exactly when the number of measurements exceeds m ≥ Const · µ 2 (U) ..."
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Cited by 124 (9 self)
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We consider the problem of reconstructing a sparse signal x 0 ∈ R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0, where U is an orthonormal matrix, we show that ℓ1 minimization recovers x 0 exactly when the number of measurements exceeds m ≥ Const · µ 2 (U) · S · log n, where S is the number of nonzero components in x 0, and µ is the largest entry in U properly normalized: µ(U) = √ n · maxk,j Uk,j. The smaller µ, the fewer samples needed. The result holds for “most ” sparse signals x 0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x 0 for each nonzero entry on T and the observed values of Ux 0 are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples.
Lassotype recovery of sparse representations for highdimensional data
 ANNALS OF STATISTICS
, 2009
"... The Lasso is an attractive technique for regularization and variable selection for highdimensional data, where the number of predictor variables pn is potentially much larger than the number of samples n. However, it was recently discovered that the sparsity pattern of the Lasso estimator can only ..."
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Cited by 122 (9 self)
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The Lasso is an attractive technique for regularization and variable selection for highdimensional data, where the number of predictor variables pn is potentially much larger than the number of samples n. However, it was recently discovered that the sparsity pattern of the Lasso estimator can only be asymptotically identical to the true sparsity pattern if the design matrix satisfies the socalled irrepresentable condition. The latter condition can easily be violated in the presence of highly correlated variables. Here we examine the behavior of the Lasso estimators if the irrepresentable condition is relaxed. Even though the Lasso cannot recover the correct sparsity pattern, we show that the estimator is still consistent in the ℓ2norm sense for fixed designs under conditions on (a) the number sn of nonzero components of the vector βn and (b) the minimal singular values of design matrices that are induced by selecting small subsets of variables. Furthermore, a rate of convergence result is obtained on the ℓ2 error with an appropriate choice of the smoothing parameter. The rate is shown to be
Robust Recovery of Signals From a Structured Union of Subspaces
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structu ..."
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Cited by 112 (43 self)
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Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a blocksparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed ℓ2/ℓ1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit, submitted
, 2007
"... Abstract. This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements – L1minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has ..."
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Cited by 102 (10 self)
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Abstract. This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements – L1minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1minimization. Our algorithm ROMP reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the Uniform Uncertainty Principle. 1.
Geometric approach to error correcting codes and reconstruction of signals
 INT. MATH. RES. NOT
, 2005
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Deterministic constructions of compressed sensing matrices
 List of References November 2001 B.1 Complete IDL Listing OMG IDL B // File: CosActivity
, 2007
"... With high esteem to Professor Henryk Wozniakowski on the occasion of his 60th birthday Compressed sensing is a new area of signal processing. Its goal is to minimize the number of samples that need to be taken from a signal for faithful reconstruction. The performance of compressed sensing on signa ..."
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Cited by 88 (2 self)
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With high esteem to Professor Henryk Wozniakowski on the occasion of his 60th birthday Compressed sensing is a new area of signal processing. Its goal is to minimize the number of samples that need to be taken from a signal for faithful reconstruction. The performance of compressed sensing on signal classes is directly related to Gelfand widths. Similar to the deeper constructions of optimal subspaces in Gelfand widths, most sampling algorithms are based on randomization. However, for possible circuit implementation, it is important to understand what can be done with purely deterministic sampling. In this note we show how to construct sampling matrices using finite fields. One such construction gives cyclic matrices which are interesting for circuit implementation. While the guaranteed performance of these deterministic constructions is not comparable to the random constructions, these matrices have the best known performance for purely deterministic constructions. 1
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 84 (21 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.