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Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case

TL;DR: In this paper, a greedy algorithm called Orthogonal Matching Pursuit (OMP) was proposed to recover a signal with m nonzero entries in dimension 1 given O(m n d) random linear measurements of that signal.
Abstract: This report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(mln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m2) measurements. The new results for OMP are comparable with recent results for another approach called Basis Pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems.

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Citations
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Journal ArticleDOI
TL;DR: An off-grid DOA estimation algorithm based on sparse Bayesian learning (SBL) is proposed in this paper, and the temporal correlation between the neighboring snapshot numbers is considered in the off- grid algorithm.
Abstract: Intelligent transportation systems (ITSs) of industrial systems have played an important role in Internet of things (IOT). The assistant calibration system (ACS) of vehicles is an emerging technology, which services the driver to drive the vehicle safely. To solve some existing problems in ACS such as frequency pairing, vehicle localization judgment, and driving in the curve road, two direction-of-arrival (DOA) estimation-based approaches are proposed to resolve these problems. However, the performance of most conventional DOA estimation algorithms is affected by the mutual coupling among the elements. The special structure of the mutual coupling matrix of the uniform linear array is applied to eliminate the effect of mutual coupling. Then, a novel on-grid DOA estimation algorithm based on compressive sensing (CS) strategies is proposed in the presence of unknown mutual coupling. In order to compensate the aperture loss of discarding information that the array receives, the array aperture is extended by the vectorization operator. In order to deal with the effect of grid mismatch, an off-grid DOA estimation algorithm based on sparse Bayesian learning (SBL) is proposed in this paper. The temporal correlation between the neighboring snapshot numbers is considered in the off-grid algorithm. The computer simulation verifies the effectiveness of the proposed algorithms.

76 citations

Journal ArticleDOI
TL;DR: A time-frequency joint sparse channel estimation for multiple-input multiple-output orthogonal frequency division multiplexing systems under the framework of structured compressive sensing (CS) demonstrates better performance and higher spectral efficiency than the conventional MIMO-OFDM schemes.
Abstract: This letter proposes a time-frequency joint sparse channel estimation for multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems under the framework of structured compressive sensing (CS). The proposed scheme first relies on a pseudorandom preamble, which is identical for all transmit antennas, to acquire the partial common support by utilizing the sparse common support property of the MIMO channels. Then, a very small amount of frequency-domain orthogonal pilots are used for the accurate channel recovery. Simulation results show that the proposed scheme demonstrates better performance and higher spectral efficiency than the conventional MIMO-OFDM schemes. Moreover, the obtained partial common support can be further utilized to reduce the complexity of the CS algorithm and improve the signal recovery probability under low signal-to-noise-ratio conditions.

76 citations

Journal ArticleDOI
TL;DR: It is shown that sparseness explicitly contributes to improved classification, hence it should not be completely ignored for computational gains and an efficient classification method is proposed based on combined representation based on dense and sparse representation.

76 citations

Journal ArticleDOI
TL;DR: A method to specifically exploit spatial sparsity property of bearing estimation algorithms by using a very small number of measurements in the form of random projections of the sensor data along with one full waveform recording at one of the sensors.
Abstract: Bearing estimation algorithms obtain only a small number of direction of arrivals (DOAs) within the entire angle domain, when the sources are spatially sparse. Hence, we propose a method to specifically exploit this spatial sparsity property. The method uses a very small number of measurements in the form of random projections of the sensor data along with one full waveform recording at one of the sensors. A basis pursuit strategy is used to formulate the problem by representing the measurements in an over complete dictionary. Sparsity is enforced by l1-norm minimization which leads to a convex optimization problem that can be efficiently solved with a linear program. This formulation is very effective for decreasing communication loads in multi sensor systems. The algorithm provides increased bearing resolution and is applicable for both narrowband and wideband signals. Sensors positions must be known, but the array shape can be arbitrary. Simulations and field data results are provided to demonstrate the performance and advantages of the proposed method.

76 citations

Journal ArticleDOI
TL;DR: CSSF MIMO radar is proposed, which applies the technique of step frequency (SF) to compressive sensing (CS) based multiple-input multiple-output (MIMO) radar, which enables high resolution range, angle and Doppler estimation, while transmitting narrowband pulses.
Abstract: A new approach is proposed, namely CSSF MIMO radar, which applies the technique of step frequency (SF) to compressive sensing (CS) based multiple-input multiple-output (MIMO) radar. The proposed approach enables high resolution range, angle and Doppler estimation, while transmitting narrowband pulses. For the case of slowly moving targets, a technique is proposed that achieves significant complexity reduction by successively estimating angle-range and Doppler in a decoupled fashion.

75 citations

References
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Book
01 Jan 1983

34,729 citations

Book
D.L. Donoho1
01 Jan 2004
TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.
Abstract: Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0

18,609 citations

Journal ArticleDOI
TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
Abstract: The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

9,950 citations

Journal ArticleDOI
TL;DR: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions, chosen in order to best match the signal structures.
Abstract: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. They derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. They compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser see (IEEE Trans. Informat. Theory, vol. 38, Mar. 1992). >

9,380 citations

Journal ArticleDOI
TL;DR: A publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates is described.
Abstract: The purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method; this connection explains the similar numerical results previously observed for the Lasso and Stagewise, and helps us understand the properties of both methods, which are seen as constrained versions of the simpler LARS algorithm. (3) A simple approximation for the degrees of freedom of a LARS estimate is available, from which we derive a Cp estimate of prediction error; this allows a principled choice among the range of possible LARS estimates. LARS and its variants are computationally efficient: the paper describes a publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates.

7,828 citations