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 optimization-based 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.

Abstract—The problem of sparsity pattern or support set recovery refers to estimating the set of nonzero coefficients of an un-3 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 non-zeroes in 3), as well as the minimum value min of 3 over its support and other parameters of measurement matrix. This paper studies the information-theoretic 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 polynomial-time Lasso method (`1-constrained quadratic programming). Index Terms—Compressed sensing, `1-relaxation, Fano’s method, high-dimensional statistical inference, information-theoretic

Abstract not found

Abstract—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 (light-tailed) 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, light-tailed environments. Index Terms—Compressed sensing (CS), impulse noise, nonlinear estimation, sampling methods, signal reconstruction. I.

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 non-adaptive 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.

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 near-optimal estimators, namely, the ML estimate in the high SNR case, and a computationally trivial thresholding algorithm in the low SNR case.

This paper studies the problem of recovering a signal with a sparse representation in a given orthonormal basis using as few noisy observations as possible. As opposed to previous studies, this paper models observations which are subject to the type of ‘clutter noise ’ encountered in radar applications (i.e., the measurements used influence the observed noise). Given this model, the paper develops bounds on the number of measurements required to reconstruct the support of the signal and the signal itself up to any given accuracy level when the measurement noise is Gaussian using non-adaptive and adaptive measurement strategies. Further, the paper demonstrates that group testing measurement constructions may be combined with statistical binary detection and estimation methods to produce practical and computationally efficient adaptive algorithms for sparse signal approximation and support recovery. In particular, the paper proves that a wide class of sparse signals can be recovered by adaptive methods using fewer noisy linear measurements than required by any recovery method based on non-adaptive Gaussian measurement ensembles. This result demonstrates an improvement over previous non-adaptive methods in the compressed sensing literature for sparse support pattern recovery in the sublinear-sparse support regime under the measurement model considered herein. 1

Compressed sensing deals with the reconstruction of sparse signals using a small number of linear measurements. One of the main challenges in compressed sensing is to find the support of a sparse signal. In the literature, several bounds on the scaling law of the number of measurements for successful support recovery have been derived where the main focus is on random Gaussian measurement matrices. In this paper, we investigate the noisy support recovery problem from an estimation theoretic point of view, where no specific assumption is made on the underlying measurement matrix. The linear measurements are perturbed by additive white Gaussian noise. We define the output of a support estimator to be a set of position values in increasing order. We set the error between the true and estimated supports as the ℓ2-norm of their difference. On the one hand, this choice allows us to use the machinery behind the ℓ2-norm error metric and on the other hand, converts the support recovery into a more intuitive and geometrical problem. First, by using the Hammersley-Chapman-Robbins (HCR) bound, we derive a fundamental lower bound on the performance of any unbiased estimator of the support set. This lower bound provides us with necessary conditions on the number of measurements for reliable ℓ2-norm support recovery, which we specifically evaluate for uniform Gaussian measurement matrices. Then, we analyze

Abstract—Consider the n-dimensional 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 maximum-likelihood estimator. We find that this probability depends (inversely) exponentially on the difference of kX k2 and the `2-norm 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 maximum-likelihood 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.

Abstract not found