Results 1  10
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16
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
 California Institute of Technology, Pasadena
, 2008
"... Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery alg ..."
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Cited by 345 (6 self)
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Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimizationbased approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(N log 2 N), where N is the length of the signal. 1.
Informationtheoretic limits on sparsity recovery in the highdimensional and noisy setting
, 2007
"... Abstract—The problem of sparsity pattern or support set recovery refers to estimating the set of nonzero coefficients of an un3 p known vector 2 based on a set of n noisy observations. It arises in a variety of settings, including subset selection in regression, graphical model selection, signal de ..."
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Cited by 51 (2 self)
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Abstract—The problem of sparsity pattern or support set recovery refers to estimating the set of nonzero coefficients of an un3 p known vector 2 based on a set of n noisy observations. It arises in a variety of settings, including subset selection in regression, graphical model selection, signal denoising, compressive sensing, and constructive approximation. The sample complexity of a given method for subset recovery refers to the scaling of the required sample size n as a function of the signal dimension p, sparsity index k (number of nonzeroes in 3), as well as the minimum value min of 3 over its support and other parameters of measurement matrix. This paper studies the informationtheoretic limits of sparsity recovery: in particular, for a noisy linear observation model based on random measurement matrices drawn from general Gaussian measurement matrices, we derive both a set of sufficient conditions for exact support recovery using an exhaustive search decoder, as well as a set of necessary conditions that any decoder, regardless of its computational complexity, must satisfy for exact support recovery. This analysis of fundamental limits complements our previous work on sharp thresholds for support set recovery over the same set of random measurement ensembles using the polynomialtime Lasso method (`1constrained quadratic programming). Index Terms—Compressed sensing, `1relaxation, Fano’s method, highdimensional statistical inference, informationtheoretic
Robust sampling and reconstruction methods for sparse signals in the precense of impulsive noise
, 2010
"... Recent results in compressed sensing show that a sparse or compressible signal can be reconstructed from a few incoherent measurements. Since noise is always present in practical data acquisition systems, sensing, and reconstruction methods are developed assuming a Gaussian (lighttailed) model for ..."
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Cited by 8 (1 self)
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Recent results in compressed sensing show that a sparse or compressible signal can be reconstructed from a few incoherent measurements. Since noise is always present in practical data acquisition systems, sensing, and reconstruction methods are developed assuming a Gaussian (lighttailed) model for the corrupting noise. However, when the underlying signal and/or the measurements are corrupted by impulsive noise, commonly employed linear sampling operators, coupled with current reconstruction algorithms, fail to recover a close approximation of the signal. In this paper, we propose robust methods for sampling and reconstructing sparse signals in the presence of impulsive noise. To solve the problem of impulsive noise embedded in the underlying signal prior the measurement process, we propose a robust nonlinear measurement operator based on the weighed myriad estimator. In addition, we introduce a geometric optimization problem based on 1 minimization employing a Lorentzian norm constraint on the residual error to recover sparse signals from noisy measurements. Analysis of the proposed methods show that in impulsive environments when the noise posses infinite variance we have a finite reconstruction error and furthermore these methods yield successful reconstruction of the desired signal. Simulations demonstrate that the proposed methods significantly outperform commonly employed compressed sensing sampling and reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, lighttailed environments.
Group Testing with Probabilistic Tests: Theory, Design and Application
"... Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a class ..."
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Cited by 7 (1 self)
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Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design nonadaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items.
A Note on Optimal Support Recovery in Compressed Sensing
, 2009
"... Recovery of the support set (or sparsity pattern) of a sparse vector from a small number of noisy linear projections (or samples) is a “compressed sensing ” problem that arises in signal processing and statistics. Although many computationally efficient recovery algorithms have been studied, the opt ..."
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Cited by 4 (0 self)
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Recovery of the support set (or sparsity pattern) of a sparse vector from a small number of noisy linear projections (or samples) is a “compressed sensing ” problem that arises in signal processing and statistics. Although many computationally efficient recovery algorithms have been studied, the optimality (or gap from optimality) of these algorithms is, in general, not well understood. In this note, approximate support recovery under a Gaussian prior is considered, and it is shown that optimal estimation depends on the recovery metric in general. By contrast, it is shown that in the SNR limits, there exist uniformly nearoptimal estimators, namely, the ML estimate in the high SNR case, and a computationally trivial thresholding algorithm in the low SNR case.
Optimal phase transitions in compressed sensing
 IEEE Trans. Inf. Theory
"... Abstract—Compressed sensing deals with efficient recovery of analog signals from linear encodings. This paper presents a statistical study of compressed sensing by modeling the input signal as an i.i.d. process with known distribution. Three classes of encoders are considered, namely optimal nonline ..."
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Cited by 3 (2 self)
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Abstract—Compressed sensing deals with efficient recovery of analog signals from linear encodings. This paper presents a statistical study of compressed sensing by modeling the input signal as an i.i.d. process with known distribution. Three classes of encoders are considered, namely optimal nonlinear, optimal linear, and random linear encoders. Focusing on optimal decoders, we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise, respectively. The optimal phasetransition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms. In particular, we show that Gaussian sensing matrices incur no penalty on the phasetransition threshold with respect to optimal nonlinear encoding. Our results also provide a rigorous justification of previous results based on replica heuristics in the weaknoise regime. Index Terms—Compressed sensing, joint sourcechannel coding, minimum meansquare error (MMSE) dimension, phase transition, random matrix, Rényi information dimension, Shannon theory.
Thresholded Basis Pursuit: Quantizing Linear Programming Solutions for Optimal Support Recovery and Approximation in Compressed Sensing. ∗
, 809
"... We consider the Compressed Sensing problem. We have a large underdetermined set of noisy measurements Y = GX + N, where X is a sparse signal and G is drawn from a random ensemble. In our previous work, we had shown that a signaltonoise ratio, SNR = O(log n) is necessary and sufficient for support ..."
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Cited by 3 (1 self)
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We consider the Compressed Sensing problem. We have a large underdetermined set of noisy measurements Y = GX + N, where X is a sparse signal and G is drawn from a random ensemble. In our previous work, we had shown that a signaltonoise ratio, SNR = O(log n) is necessary and sufficient for support recovery from an informationtheoretic perspective. In this paper we present a linear programming solution for support recovery. The solution of the problem amounts to solving min ‖Z‖1 s.t. Y = GZ, and quantizing/thresholding the resulting solution Z. We show that this scheme is guaranteed to perfectly reconstruct a discrete signal or control the elementwise reconstruction error for a continuous signal for specific values of sparsity. We show that in the linear regime when the sparsity, k, increases linearly with signal dimension, n, the sign pattern of X can be recovered with SNR = O(log n) and m = O(k) measurements. Our proof technique is based on perturbation of the noiseless ℓ1 problem. Consequently, the achievable sparsity level in the noisy problem is comparable to that of the noiseless problem. Our result offers a sharp characterization in that neither the SNR nor the sparsity ratio can be significantly improved. In contrast previous results based on LASSO and MAXCorrelation techniques assume significantly larger SNR or sublinear sparsity. We also show that our final result can be obtained from Dvoretsky theorem rather than the restricted isometry property (RIP). The advantage of this line of reasoning is that Dvoretsky’s theorem continues to hold for nonsingular transformations while RIP property may not be satisfied for the latter case. We also consider approximation in terms of ℓ2 and show that our bounds match existing bounds for LASSO in this case. 1
Nearly Sharp Sufficient Conditions on Exact Sparsity Pattern Recovery
"... Abstract—Consider the ndimensional vector y = X + where 2 p has only k nonzero entries and 2 n is a Gaussian noise. This can be viewed as a linear system with sparsity constraints corrupted by noise, where the objective is to estimate the sparsity pattern of given the observation vector y and the m ..."
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Cited by 2 (0 self)
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Abstract—Consider the ndimensional vector y = X + where 2 p has only k nonzero entries and 2 n is a Gaussian noise. This can be viewed as a linear system with sparsity constraints corrupted by noise, where the objective is to estimate the sparsity pattern of given the observation vector y and the measurement matrix X. First, we derive a nonasymptotic upper bound on the probability that a specific wrong sparsity pattern is identified by the maximumlikelihood estimator. We find that this probability depends (inversely) exponentially on the difference of kX k2 and the `2norm of X projected onto the range of columns of X indexed by the wrong sparsity pattern. Second, when X is randomly drawn from a Gaussian ensemble, we calculate a nonasymptotic upper bound on the probability of the maximumlikelihood decoder not declaring (partially) the true sparsity pattern. Consequently, we obtain sufficient conditions on the sample size n that guarantee almost surely the recovery of the true sparsity pattern. We find that the required growth rate of sample size n matches the growth rate of previously established necessary conditions. Index Terms—Hypothesis testing, random projections, sparsity pattern recovery, subset selection, underdetermined systems of equations. I.