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.
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TL;DR: Simulation results show that using the pilot patterns designed by the two proposed methods for the CS-based channel estimation in MIMO-OFDM systems gives a much better performance than using other pilot patterns in terms of the mean square error of the channel estimate as well as the bit error rate of the system.
Abstract: The frequency-selective channel-estimation problem in multi-input-multi-output orthogonal frequency division multiplexing (MIMO-OFDM) systems is investigated from the perspective of compressed sensing (CS). By minimizing the mutual coherence of the measurement matrix in CS theory, two pilot allocation methods for the CS-based channel estimation in MIMO-OFDM systems are proposed. Simulation results show that using the pilot patterns designed by the two proposed methods gives a much better performance than using other pilot patterns in terms of the mean square error of the channel estimate as well as the bit error rate of the system. Moreover, the optimal pilot patterns obtained by the proposed second method based on genetic algorithm and shift mechanism could offer a larger performance gain than those by the first method based on minimizing the largest element in the mutual coherence set possessed by pilot patterns for all multiple antenna ports.
65 citations
TL;DR: A new Cluster Size Load Balancing for CS algorithm (CSLB-CS) is proposed which could achieve optimal utilization of CS method in an IoT-based sensor network and exceeds the performance of hybrid CS and plain CS in terms of network lifetime, overall energy consumption, total number of data transmitted, and reconstruction error.
64 citations
28 Jun 2016
TL;DR: The algorithm MetaPalette is presented, which uses long k-mer sizes to fit a k-Mer “palette” of a given sample to the k-MER palette of reference organisms, and returns a traditional, fixed-rank taxonomic profile which is shown on independently simulated data to be one of the most accurate to date.
Abstract: Metagenomic profiling is challenging in part because of the highly uneven sampling of the tree of life by genome sequencing projects and the limitations imposed by performing phylogenetic inference at fixed taxonomic ranks. We present the algorithm MetaPalette, which uses long k -mer sizes ( k = 30, 50) to fit a k -mer “palette” of a given sample to the k -mer palette of reference organisms. By modeling the k -mer palettes of unknown organisms, the method also gives an indication of the presence, abundance, and evolutionary relatedness of novel organisms present in the sample. The method returns a traditional, fixed-rank taxonomic profile which is shown on independently simulated data to be one of the most accurate to date. Tree figures are also returned that quantify the relatedness of novel organisms to reference sequences, and the accuracy of such figures is demonstrated on simulated spike-ins and a metagenomic soil sample. The software implementing MetaPalette is available at: https://github.com/dkoslicki/MetaPalette. Pretrained databases are included for Archaea , Bacteria , Eukaryota , and viruses. IMPORTANCE Taxonomic profiling is a challenging first step when analyzing a metagenomic sample. This work presents a method that facilitates fine-scale characterization of the presence, abundance, and evolutionary relatedness of organisms present in a given sample but absent from the training database. We calculate a “ k -mer palette” which summarizes the information from all reads, not just those in conserved genes or containing taxon-specific markers. The compositions of palettes are easy to model, allowing rapid inference of community composition. In addition to providing strain-level information where applicable, our approach provides taxonomic profiles that are more accurate than those of competing methods. Author Video : An author video summary of this article is available.
64 citations
Posted Content•
TL;DR: In this paper, the authors considered the block length as a parameter of the system and established sharp lower bounds on the allowable block-sparsity as a function of the block-length.
Abstract: One of the most basic problems in compressed sensing is solving an under-determined system of linear equations. Although this problem seems rather hard certain l1-optimization algorithm appears to be very successful in solving it. The recent work of [14, 28] rigorously proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that l1-optimization algorithm succeeds in solving the system. In more recent papers [78,81] we considered the setup of the so-called block-sparse unknown vectors. In a large dimensional and statistical context, we determined sharp lower bounds on the values of allowable sparsity for any given number (proportional to the length of the unknown vector) of equations such that an l2/l1-optimization algorithm succeeds in solving the system. The results established in [78, 81] assumed a fairly large block-length of the block-sparse vectors. In this paper we consider the block-length to be a parameter of the system. Consequently, we then establish sharp lower bounds on the values of the allowable block-sparsity as functions of the block-length.
64 citations
TL;DR: In this article, a bi-level protected compressive sampling (BLP-CS) model is proposed, which makes use of the advantage of the non-RIP measurement matrix construction.
Abstract: Some pioneering works have investigated embedding cryptographic properties in compressive sampling (CS) in a way similar to one-time pad symmetric cipher. This paper tackles the problem of constructing a CS-based symmetric cipher under the key reuse circumstance, i.e., the cipher is resistant to common attacks even a fixed measurement matrix is used multiple times. To this end, we suggest a bi-level protected CS (BLP-CS) model which makes use of the advantage of the non-RIP measurement matrix construction. Specifically, two kinds of artificial basis mismatch techniques are investigated to construct key-related sparsifying bases. It is demonstrated that the encoding process of BLP-CS is simply a random linear projection, which is the same as the basic CS model. However, decoding the linear measurements requires knowledge of both the key-dependent sensing matrix and its sparsifying basis. The proposed model is exemplified by sampling images as a joint data acquisition and protection layer for resource-limited wireless sensors. Simulation results and numerical analyses have justified that the new model can be applied in circumstances where the measurement matrix can be re-used.
64 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