Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case
01 Aug 2007-
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.
Citations
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TL;DR: This work proposes to use structured output regression machine (SORM) to simultaneously model the inherent spatial relations between the HR and LR patches and introduces a nonlocal (NL) self-similarity prior in natural images to further enhance the SORM-based SR results.
Abstract: For regression-based single-image super-resolution (SR) problem, the key is to establish a mapping relation between high-resolution (HR) and low-resolution (LR) image patches for obtaining a visually pleasing quality image. Most existing approaches typically solve it by dividing the model into several single-output regression problems, which obviously ignores the circumstance that a pixel within an HR patch affects other spatially adjacent pixels during the training process, and thus tends to generate serious ringing artifacts in resultant HR image as well as increase computational burden. To alleviate these problems, we propose to use structured output regression machine (SORM) to simultaneously model the inherent spatial relations between the HR and LR patches, which is propitious to preserve sharp edges. In addition, to further improve the quality of reconstructed HR images, a nonlocal (NL) self-similarity prior in natural images is introduced to formulate as a regularization term to further enhance the SORM-based SR results. To offer a computation-effective SORM method, we use a relative small nonsupport vector samples to establish the accurate regression model and an accelerating algorithm for NL self-similarity calculation. Extensive SR experiments on various images indicate that the proposed method can achieve more promising performance than the other state-of-the-art SR methods in terms of both visual quality and computational cost.
48 citations
TL;DR: Simulation results show that the proposed channel estimation algorithm, with the new pilot placement scheme, offers a better tradeoff between the performance-in terms of mean-squared-error (MSE) and bit-error-rate (BER)-and complexity, when compared to other estimation algorithms.
Abstract: This paper presents a novel recovery algorithm based on sparsity adaptive matching pursuit (SaMP) and a new near-optimal pilot placement scheme, for compressed sensing (CS)-based sparse channel estimation in orthogonal frequency division multiplexing (OFDM) communication systems. Compared with other state-of-the-art recovery algorithms, the proposed algorithm possesses the feature of SaMP of not requiring a priori knowledge of the sparsity level, and moreover, adjusts the step size adaptively to approach the true sparsity level. Furthermore, we focus on the pilot pattern design in sparse channel estimation. Although a brute-force search guarantees the optimal pilot pattern, it is prohibitive to examine all possibilities due to high computational complexity. It is known that by minimizing the mutual coherence of the measurement matrix when the signal is sparse on the unitary discrete Fourier transform (DFT) matrix, the optimal set of pilot locations is a cyclic difference set (CDS). Based on this, we propose an efficient near-optimal pilot placement scheme in cases where CDS does not exist. Simulation results show that the proposed channel estimation algorithm, with the new pilot placement scheme, offers a better tradeoff between the performance—in terms of mean-squared-error (MSE) and bit-error-rate (BER)—and complexity, when compared to other estimation algorithms.
48 citations
TL;DR: This analysis shows that OMP can accurately recover all K-sparse signals within [2.8 K] iterations when the measurement matrix satisfies a restricted isometry property (RIP).
Abstract: Orthogonal matching pursuit (OMP) is a greedy algorithm widely used for the recovery of sparse signals from compressed measurements. In this paper, we analyze the number of iterations required for the OMP algorithm to perform exact recovery of sparse signals. Our analysis shows that OMP can accurately recover all $K$ -sparse signals within $\lceil 2.8 \; K \rceil$ iterations when the measurement matrix satisfies a restricted isometry property (RIP). Our result improves upon the recent result of Zhang and also bridges the gap between Zhang's result and the fundamental limit of OMP at which exact recovery of $K$ -sparse signals cannot be uniformly guaranteed.
48 citations
13 Apr 2009
TL;DR: A Bayesian framework for the localization problem is introduced and it is demonstrated that the optimal solution can be computed using fast sparse approximation algorithms, and that exploiting the signal sparsity can reduce the sensing and computational cost on the sensors, as well as the communication bandwidth.
Abstract: This paper exploits recent developments in sparse approximation and compressive sensing to efficiently perform localization in a sensor network. We introduce a Bayesian framework for the localization problem and provide sparse approximations to its optimal solution. By exploiting the spatial sparsity of the posterior density, we demonstrate that the optimal solution can be computed using fast sparse approximation algorithms. We show that exploiting the signal sparsity can reduce the sensing and computational cost on the sensors, as well as the communication bandwidth. We further illustrate that the sparsity of the source locations can be exploited to decentralize the computation of the source locations and reduce the sensor communications even further. We also discuss how recent results in 1-bit compressive sensing can significantly reduce the amount of inter-sensor communications by transmitting only the intrinsic timing information. Finally, we develop a computationally efficient algorithm for bearing estimation using a network of sensors with provable guarantees.
48 citations
TL;DR: A probabilistic model for sparse signal reconstruction is proposed and several novel algorithms for computing the maximum likelihood (ML) parameter estimates under this model are developed.
Abstract: We propose a probabilistic model for sparse signal reconstruction and develop several novel algorithms for computing the maximum likelihood (ML) parameter estimates under this model. The measurements follow an underdetermined linear model where the regression-coefficient vector is the sum of an unknown deterministic sparse signal component and a zero-mean white Gaussian component with an unknown variance. Our reconstruction schemes are based on an expectation-conditional maximization either (ECME) iteration that aims at maximizing the likelihood function with respect to the unknown parameters for a given signal sparsity level. Compared with the existing iterative hard thresholding (IHT) method, the ECME algorithm contains an additional multiplicative term and guarantees monotonic convergence for a wide range of sensing (regression) matrices. We propose a double overrelaxation (DORE) thresholding scheme for accelerating the ECME iteration. We prove that, under certain mild conditions, the ECME and DORE iterations converge to local maxima of the likelihood function. The ECME and DORE iterations can be implemented exactly in small-scale applications and for the important class of large-scale sensing operators with orthonormal rows used e.g., partial fast Fourier transform (FFT). If the signal sparsity level is unknown, we introduce an unconstrained sparsity selection (USS) criterion and a tuning-free automatic double overrelaxation (ADORE) thresholding method that employs USS to estimate the sparsity level. We compare the proposed and existing sparse signal reconstruction methods via one-dimensional simulation and two-dimensional image reconstruction experiments using simulated and real X-ray CT data.
48 citations
References
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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
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
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
Stanford University1, Cleveland Clinic2, University of Toronto3, Centre national de la recherche scientifique4, Université Paris-Saclay5, University of Paris-Sud6, Avaya7, Rutgers University8, RAND Corporation9, IBM10, University of Pennsylvania11, University of Western Australia12, University of Minnesota13
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