Recovery of an unknown sparse signal from a few of its projections is the key objective of compressed sensing. Often one comes across signals that are not ordinarily sparse but are sparse blockwise. Existing block sparse recovery algorithms like BOMP make the assumption of uniform block size and known block boundaries, which are, however, not very practical in many applications. This paper addresses this problem and proposes a two step procedure, where the first stage is a coarse block location identification stage while the second stage carries out finer localization of a non-zero cluster within the window selected in the first stage. A detailed convergence analysis of the proposed algorithm is carried out by first defining the so-called pseudoblock-interleaved block RIP of the given generalized block sparse signal and then imposing upper bounds on the corresponding RIC. We also extend the analysis for complex vector as well as matrix entries where it turns out that the extension is non-trivial and requires special care. Furthermore, assuming real Gaussian sensing matrix entries, we find a lower bound on the probability that the derived recovery bounds are satisfied. The lower bound suggests that there are sets of parameters such that the derived bound is satisfied with high probability. Simulation results confirm significantly improved performance of the proposed algorithm as compared to BOMP.
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Univ. Paris-Sud（巴黎南大学）；Université Paris-Saclay（巴黎萨克雷大学）
Greedy algorithms for feature selection are widely used for recovering sparse high-dimensional vectors in linear models. In classical procedures, the main emphasis was put on the sample complexity, with little or no consideration of the computation resources required. We present a novel online algorithm: Online Orthogonal Matching Pursuit (OOMP) for online support recovery in the random design setting of sparse linear regression. Our procedure selects features sequentially, alternating between allocation of samples only as needed to candidate features, and optimization over the selected set of…
Subspace clustering methods based on $\ell_1$, $\ell_2$ or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions (e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, $\ell_2…
Texas A&M University（德克萨斯A&M大学）；Cainiao Network（菜鸟网络）；University of California, Davis（加州大学戴维斯分校）；The Chinese University of Hong Kong（香港中文大学）
We consider the problem of maximizing the $\ell_1$ norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is numerically challenging due to its nonsmooth objective and nonconvex constraint, and its algorithmic aspects have not been well explored. In this paper, we show how the manifold structure of the sphere can be exploited to design fast algorithms for tackling this problem. Specifically, our contribution is threefold. First, we present a manifold proximal point …
Georgia Institute of Technology（佐治亚理工学院）；Zuse Institute Berlin（柏林祖斯研究所）；Technische Universität Berlin（柏林工业大学）
Matching pursuit algorithms are an important class of algorithms in signal processing and machine learning. We present a blended matching pursuit algorithm, combining coordinate descent-like steps with stronger gradient descent steps, for minimizing a smooth convex function over a linear space spanned by a set of atoms. We derive sublinear to linear convergence rates according to the smoothness and sharpness orders of the function and demonstrate computational superiority of our approach. In particular, we derive linear rates for a wide class of non-strongly convex functions, and we demonstrat…
In this paper, we propose a new orthogonal matching pursuit algorithm called quasi-OMP algorithm which greatly enhances the performance of classical orthogonal matching pursuit (OMP) algorithm, at some cost of computational complexity. We are able to show that under some sufficient conditions of mutual coherence of the sensing matrix, the QOMP Algorithm succeeds in recovering the s-sparse signal vector x within s iterations where a total number of 2s columns are selected under the both noiseless and noisy settings. In addition, we show that for Gaussian sensing matrix, the norm of the residual…
The orthogonal matching pursuit (OMP) is one of the mainstream algorithms for sparse data reconstruction or approximation. It acts as a driving force for the development of several other greedy methods for sparse data reconstruction, and it also plays a vital role in the development of compressed sensing theory for sparse signal and image reconstruction. In this paper, we propose the so-called dynamic orthogonal matching pursuit (DOMP) and enhanced dynamic orthogonal matching pursuit (EDOMP) algorithms which are more efficient than OMP for sparse data reconstruction from a numerical point of v…
Greed is good. However, the tighter you squeeze, the less you have. In this paper, a less greedy algorithm for sparse signal reconstruction in compressive sensing, named orthogonal matching pursuit with thresholding is studied. Using the global 2-coherence , which provides a "bridge" between the well known mutual coherence and the restricted isometry constant, the performance of orthogonal matching pursuit with thresholding is analyzed and more general results for sparse signal reconstruction are obtained. It is also shown that given the same assumption on the coherence index and the restricte…
The Orthogonal Matching Pursuit (OMP) for compressed sensing iterates over a scheme of support augmentation and signal estimation. We present two novel matching pursuit algorithms with intrinsic regularization of the signal estimation step that do not rely on a priori knowledge of the signal's sparsity. An iterative approach allows for a hardware efficient implementation of our algorithm, and enables real-world applications of compressed sensing. We provide a series of numerical examples that demonstrate a good performance, especially when the number of measurements is relatively small.
Several exact recovery criteria (ERC) ensuring that orthogonal matching pursuit (OMP) identifies the correct support of sparse signals have been developed in the last few years. These ERC rely on the restricted isometry property (RIP), the associated restricted isometry constant (RIC) and sometimes the restricted orthogonality constant (ROC). In this paper, three of the most recent ERC for OMP are examined. The contribution is to show that these ERC remain valid for a generalization of OMP, entitled simultaneous orthogonal matching pursuit (SOMP), that is capable to process several measurement…
This paper provides a simple proof of the mutual incoherence condition $μ< \frac{1}{2K-1}$ under which K-sparse signal can be accurately reconstructed from a small number of linear measurements using the orthogonal matching pursuit (OMP) algorithm. Our proof, based on mathematical induction, is built on an observation that the general step of the OMP process is in essence same as the initial step since the residual is considered as a new measurement preserving the sparsity level of an input vector.
In this paper, we consider orthogonal matching pursuit (OMP) algorithm for multiple measurement vectors (MMV) problem. The robustness of OMPMMV is studied under general perturbations---when the measurement vectors as well as the sensing matrix are incorporated with additive noise. The main result shows that although exact recovery of the sparse solutions is unrealistic in noisy scenario, recovery of the support set of the solutions is guaranteed under suitable conditions. Specifically, a sufficient condition is derived that guarantees exact recovery of the sparse solutions in noiseless scenari…
In this paper we present a new coherence-based performance guarantee for the Orthogonal Matching Pursuit (OMP) algorithm. An upper bound for the probability of correctly identifying the support of a sparse signal with additive white Gaussian noise is derived. Compared to previous work, the new bound takes into account the signal parameters such as dynamic range, noise variance, and sparsity. Numerical simulations show significant improvements over previous work.
In this paper we define a new coherence index, named the global 2-coherence, of a given dictionary and study its relationship with the traditional mutual coherence and the restricted isometry constant. By exploring this relationship, we obtain more general results on sparse signal reconstruction using greedy algorithms in the compressive sensing (CS) framework. In particular, we obtain an improved bound over the best known results on the restricted isometry constant for successful recovery of sparse signals using orthogonal matching pursuit (OMP).
This paper demonstrates theoretically that if the restricted isometry constant $δ_K$ of the compressed sensing matrix satisfies $$ δ_{K+1} < \frac{1}{\sqrt{K}+1}, $$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover a signal with $K$ nonzero entries in $K$ iterations. In contrast, matrices are also constructed with restricted isometry constant $$ δ_{K+1} = \frac{1}{\sqrt{K}} $$ such that OMP can not recover $K$-sparse $x$ in $K$ iterations. This result shows that the conjecture given by Dai and Milenkovic is ture.
This paper demonstrates that if the restricted isometry constant $δ_{K+1}$ of the measurement matrix $A$ satisfies $$ δ_{K+1} < \frac{1}{\sqrt{K}+1}, $$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every $K$--sparse signal $\mathbf{x}$ in $K$ iterations from $A\x$. By contrast, a matrix is also constructed with the restricted isometry constant $$ δ_{K+1} = \frac{1}{\sqrt{K}} $$ such that OMP can not recover some $K$-sparse signal $\mathbf{x}$ in $K$ iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009.
Sampling rate is the bottleneck for spectrum sensing over multi-GHz bandwidth. Recent progress in compressed sensing (CS) initialized several sub-Nyquist rate approaches to overcome the problem. However, efforts to design CS reconstruction algorithms for wideband spectrum sensing are very limited. It is possible to further reduce the sampling rate requirement and improve reconstruction performance via algorithms considering prior knowledge of cognitive radio spectrum usages. In this paper, we group the usages of cognitive radio spectrum into three categories and propose a modified orthogonal m…
This paper presents a new analysis for the orthogonal matching pursuit (OMP) algorithm. It is shown that if the restricted isometry property (RIP) is satisfied at sparsity level $O(\bar{k})$, then OMP can recover a $\bar{k}$-sparse signal in 2-norm. For compressed sensing applications, this result implies that in order to uniformly recover a $\bar{k}$-sparse signal in $\Real^d$, only $O(\bar{k} \ln d)$ random projections are needed. This analysis improves earlier results on OMP that depend on stronger conditions such as mutual incoherence that can only be satisfied with $Ω(\bar{k}^2 \ln d)$ ra…
A recursive approach for shrinking coefficients of an atomic decomposition is proposed. The corresponding algorithm evolves so as to provide at each iteration a) the orthogonal projection of a signal onto a reduced subspace and b) the index of the coefficient to be disregarded in order to construct a coarser approximation minimizing the norm of the residual error.
In this paper we present a new coherence-based performance guarantee for the Orthogonal Matching Pursuit (OMP) algorithm. A lower bound for the probability of correctly identifying the support of a sparse signal with additive white Gaussian noise is derived. Compared to previous work, the new bound takes into account the signal parameters such as dynamic range, noise variance, and sparsity. Numerical simulations show significant improvements over previous work and a closer match to empirically obtained results of the OMP algorithm.
Sparse Subspace Clustering (SSC) is a state-of-the-art method for clustering high-dimensional data points lying in a union of low-dimensional subspaces. However, while $\ell_1$ optimization-based SSC algorithms suffer from high computational complexity, other variants of SSC, such as Orthogonal Matching Pursuit-based SSC (OMP-SSC), lose clustering accuracy in pursuit of improving time efficiency. In this letter, we propose a novel Active OMP-SSC, which improves clustering accuracy of OMP-SSC by adaptively updating data points and randomly dropping data points in the OMP process, while still en…
A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations is that the restricted isometry constant of the sensing matrix satisfies delta_K+1<1/(sqrt(delta_K+1)+1). By exploiting an approximate orthogonality condition characterized via the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the upper bound on delta can be further relaxed to delta_K+1<(sqrt(1+4*delta_K+1)-1)/(2K).This result thus narrows the gap between the so far best known bound and the ultimate perfo…
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 re…
Compressed sensing is a developing field aiming at reconstruction of sparse signals acquired in reduced dimensions, which make the recovery process under-determined. The required solution is the one with minimum $\ell_0$ norm due to sparsity, however it is not practical to solve the $\ell_0$ minimization problem. Commonly used techniques include $\ell_1$ minimization, such as Basis Pursuit (BP) and greedy pursuit algorithms such as Orthogonal Matching Pursuit (OMP) and Subspace Pursuit (SP). This manuscript proposes a novel semi-greedy recovery approach, namely A* Orthogonal Matching Pursuit (…
We study the problem of recovering the sparsity pattern of block-sparse signals from noise-corrupted measurements. A simple, efficient recovery method, namely, a block-version of the orthogonal matching pursuit (OMP) method, is considered in this paper and its behavior for recovering the block-sparsity pattern is analyzed. We provide sufficient conditions under which the block-version of the OMP can successfully recover the block-sparse representations in the presence of noise. Our analysis reveals that exploiting block-sparsity can improve the recovery ability and lead to a guaranteed recover…
We shall show that if the restricted isometry constant (RIC) $δ_{s+1}(A)$ of the measurement matrix $A$ satisfies $$ δ_{s+1}(A) < \frac{1}{\sqrt{s + 1}}, $$ then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every $s$-sparse signal $x$ in $s$ iterations from $b = Ax$. Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given $s \in \N$, we shall construct a matrix $A$ with the RIC $$ δ_{s+1}(A) = \frac{1}{\sqrt{s + 1}} $$ such that OMP may not recover some $s$-sparse signal $x$ in $s$ iterations.
We present a theoretical analysis of the average performance of OMP for sparse approximation. For signals that are generated from a dictionary with $K$ atoms and coherence $μ$ and coefficients corresponding to a geometric sequence with parameter $α<1$, we show that OMP is successful with high probability as long as the sparsity level $S$ scales as $Sμ^2 \log K \lesssim 1-α$. This improves by an order of magnitude over worst case results and shows that OMP and its famous competitor Basis Pursuit outperform each other depending on the setting.
Orthogonal matching pursuit (OMP) is a widely used algorithm for recovering sparse high dimensional vectors in linear regression models. The optimal performance of OMP requires \textit{a priori} knowledge of either the sparsity of regression vector or noise statistics. Both these statistics are rarely known \textit{a priori} and are very difficult to estimate. In this paper, we present a novel technique called residual ratio thresholding (RRT) to operate OMP without any \textit{a priori} knowledge of sparsity and noise statistics and establish finite sample and large sample support recovery gu…
Compressed Sensing (CS) is a signal processing technique which can accurately recover sparse signals from linear measurements with far fewer number of measurements than those required by the classical Shannon-Nyquist theorem. Block sparse signals, i.e., the sparse signals whose nonzero coefficients occur in few blocks, arise from many fields. Block orthogonal matching pursuit (BOMP) is a popular greedy algorithm for recovering block sparse signals due to its high efficiency and effectiveness. By fully using the block sparsity of block sparse signals, BOMP can achieve very good recovery perform…
The simultaneous orthogonal matching pursuit (SOMP) algorithm aims to find the joint support of a set of sparse signals acquired under a multiple measurement vector model. Critically, the analysis of SOMP depends on the maximal inner product of any atom of a suitable dictionary and the current signal residual, which is formed by the subtraction of previously selected atoms. This inner product, or correlation, is a key metric to determine the best atom to pick at each iteration. This paper provides, for each iteration of SOMP, a novel lower bound of the aforementioned metric for the atoms belon…
This paper considers the problem of tracking a dynamic sparse channel in a broadband wireless communication system. A probabilistic signal model is firstly proposed to describe the special features of temporal correlations of dynamic sparse channels: path delays change slowly over time, while path gains evolve faster. Based on such temporal correlations, we then propose the differential orthogonal matching pursuit (D-OMP) algorithm to track a dynamic sparse channel in a sequential way by updating the small channel variation over time. Compared with other channel tracking algorithms, simulation…
Sparsely-Packetized Predictive Control by Orthogonal Matching Pursuit
Graduate School of Informatics, Kyoto University（京都大学情报学研究科）；School of Electrical Engineering & Computer Science, The University of Newcastle（纽卡斯尔大学电气与计算机工程学院）；Department of Electronic Systems, Aalborg University（奥尔堡大学电子系统系）
Direction of Arrival (DOA) estimation of multiple narrow-band coherent or partially coherent sources is a major challenge in array signal processing. Though many subspace- based algorithms are available in literature, none of them tackle the problem of resolving coherent sources directly, e.g. without modifying the sample data covariance matrix. Compressive Sensing (CS) based sparse recovery algorithms are being applied as a novel technique to this area. In this paper, we introduce Orthogonal Matching Pursuit (OMP) to the DOA estimation problem. We demonstrate how a DOA estimation problem can …
We consider the block orthogonal multi-matching pursuit (BOMMP) algorithm for the recovery of block sparse signals. A sharp bound is obtained for the exact reconstruction of block $K$-sparse signals via the BOMMP algorithm in the noiseless case, based on the block restricted isometry constant (block-RIC). Moreover, we show that the sharp bound combining with an extra condition on the minimum $\ell_2$ norm of nonzero blocks of block $K-$sparse signals is sufficient to recover the true support of block $K$-sparse signals by the BOMMP in the noise case. The significance of the results we obtain i…
In this paper, multi-snapshot Newtonized orthogonal matching pursuit (MNOMP) algorithm is proposed to deal with the line spectrum estimation with multiple measurement vectors (MMVs). MNOMP has the low computation complexity and state-of-the-art performance advantage of NOMP, and also includes two key steps: Detecting a new sinusoid on an oversampled discrete Fourier transform (DFT) grid and refining the parameters of already detected sinusoids to avoid the problem of basis mismatch. We provide a stopping criterion based on the overestimating probability of the model order. In addition, the con…
Generalized Orthogonal Matching Pursuit (gOMP) is a natural extension of OMP algorithm where unlike OMP, it may select $N (\geq1)$ atoms in each iteration. In this paper, we demonstrate that gOMP can successfully reconstruct a $K$-sparse signal from a compressed measurement $ {\bf y}={\bf Φx}$ by $K^{th}$ iteration if the sensing matrix ${\bf Φ}$ satisfies restricted isometry property (RIP) of order $NK$ where $δ_{NK} < \frac {\sqrt{N}}{\sqrt{K}+2\sqrt{N}}$. Our bound offers an improvement over the very recent result shown in \cite{wang_2012b}. Moreover, we present another bound for gOMP of or…
Generalized orthogonal matching pursuit (gOMP), also called orthogonal multi-matching pursuit, is an extension of OMP in the sense that $N\geq1$ indices are identified per iteration. In this paper, we show that if the restricted isometry constant (RIC) $δ_{NK+1}$ of a sensing matrix $\A$ satisfies $δ_{NK+1} < 1/\sqrt {K/N+1}$, then under a condition on the signal-to-noise ratio, gOMP identifies at least one index in the support of any $K$-sparse signal $\x$ from $\y=\A\x+\v$ at each iteration, where $\v$ is a noise vector. Surprisingly, this condition does not require $N\leq K$ which is needed…
A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from 4 k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n approaches infinity. This work strengthens this result by showing that a lower number of measurements, 2 k log(n - k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kmin <= k <= kmax but is unknown, 2 kmax log(n - kmin) measurements is sufficient. Furthermore, this n…
Orthogonal Matching Pursuit (OMP) is a canonical greedy pursuit algorithm for sparse approximation. Previous studies of OMP have mainly considered the exact recovery of a sparse signal $\bm x$ through $\bm Φ$ and $\bm y=\bm Φ\bm x$, where $\bm Φ$ is a matrix with more columns than rows. In this paper, based on Restricted Isometry Property (RIP), the performance of OMP is analyzed under general perturbations, which means both $\bm y$ and $\bm Φ$ are perturbed. Though exact recovery of an almost sparse signal $\bm x$ is no longer feasible, the main contribution reveals that the exact recovery of…
As a greedy algorithm to recover sparse signals from compressed measurements, orthogonal matching pursuit (OMP) algorithm has received much attention in recent years. In this paper, we introduce an extension of the OMP for pursuing efficiency in reconstructing sparse signals. Our approach, henceforth referred to as generalized OMP (gOMP), is literally a generalization of the OMP in the sense that multiple $N$ indices are identified per iteration. Owing to the selection of multiple ''correct'' indices, the gOMP algorithm is finished with much smaller number of iterations when compared to the OM…
Generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of orthogonal matching pursuit. It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every $K$-sparse signal via the gOMP algorithm in the noiseless case. That is, if the restricted isometry constant (RIC) $δ_{NK+1}$ of the sensing matrix $A$ satisfies \begin{eqnarray*} δ_{NK+1}<\frac{1}{\sqrt{\frac{K}{N}+1}}, \end{eqnarray*} then the gOMP can perfectly recover every $K$-sparse signal $x$ from …
The sixth-generation networks envision the terahertz (THz) band as one of the key enabling technologies because of its ultrawide bandwidth. To combat the severe attenuation, the THz wireless systems employ large arrays, wherein the near-field beam-split (NB) severely degrades the accuracy of channel acquisition. Contrary to prior works that examine only either narrowband beamforming or far-field models, we estimate the wideband THz channel via an NB-aware orthogonal matching pursuit (NBA-OMP) approach. We design an NBA dictionary of near-field steering vectors by exploiting the corresponding a…
This paper studies the joint support recovery of similar sparse vectors on the basis of a limited number of noisy linear measurements, i.e., in a multiple measurement vector (MMV) model. The additive noise signals on each measurement vector are assumed to be Gaussian and to exhibit different variances. The simultaneous orthogonal matching pursuit (SOMP) algorithm is generalized to weight the impact of each measurement vector on the choice of the atoms to be picked according to their noise levels. The new algorithm is referred to as SOMP-NS where NS stands for noise stabilization. To begin with…
In this letter, we first construct a counter example to show that for any given positive integer $K\geq 2$ and for any $\frac{1}{\sqrt{K+1}}\leq t<1$, there always exist a $K-$sparse $\x$ and a matrix $\A$ with the restricted isometry constant $δ_{K+1}=t$ such that the OMP algorithm fails in $K$ iterations. Secondly, we show that even when $δ_{K+1}=\frac{1}{\sqrt{K}+1}$, the OMP algorithm can also perfectly recover every $K-$sparse vector $\x$ from $\y=\A\x$ in $K$ iteration. This improves the best existing results which were independently given by Mo et al. and Wang et al.
The orthogonal matching pursuit (OMP) algorithm is a commonly used algorithm for recovering $K$-sparse signals $\x\in \mathbb{R}^{n}$ from linear model $\y=\A\x$, where $\A\in \mathbb{R}^{m\times n}$ is a sensing matrix. A fundamental question in the performance analysis of OMP is the characterization of the probability that it can exactly recover $\x$ for random matrix $\A$. Although in many practical applications, in addition to the sparsity, $\x$ usually also has some additional property (for example, the nonzero entries of $\x$ independently and identically follow the Gaussian distribution…
We propose a fast sequential algorithm for the fundamental problem of estimating frequencies and amplitudes of a noisy mixture of sinusoids. The algorithm is a natural generalization of Orthogonal Matching Pursuit (OMP) to the continuum using Newton refinements, and hence is termed Newtonized OMP (NOMP). Each iteration consists of two phases: detection of a new sinusoid, and sequential Newton refinements of the parameters of already detected sinusoids. The refinements play a critical role in two ways: (1) sidestepping the potential basis mismatch from discretizing a continuous parameter space,…
As an extension of orthogonal matching pursuit (OMP) improving the recovery performance of sparse signals, generalized OMP (gOMP) has recently been studied in the literature. In this paper, we present a new analysis of the gOMP algorithm using restricted isometry property (RIP). We show that if the measurement matrix $\mathbfΦ \in \mathcal{R}^{m \times n}$ satisfies the RIP with $$δ_{\max \left\{9, S + 1 \right\}K} \leq \frac{1}{8},$$ then gOMP performs stable reconstruction of all $K$-sparse signals $\mathbf{x} \in \mathcal{R}^n$ from the noisy measurements $\mathbf{y} = \mathbf{Φx} + \mathbf…
In this paper, the joint support recovery of several sparse signals whose supports present similarities is examined. Each sparse signal is acquired using the same noisy linear measurement process, which returns fewer observations than the dimension of the sparse signals. The measurement noise is assumed additive, Gaussian, and admits different variances for each sparse signal that is measured. Using the theory of compressed sensing, the performance of simultaneous orthogonal matching pursuit (SOMP) is analysed for the envisioned signal model. The cornerstone of this paper is a novel analysis m…
Support recovery of sparse signals from compressed linear measurements is a fundamental problem in compressed sensing (CS). In this paper, we study the orthogonal matching pursuit (OMP) algorithm for the recovery of support under noise. We consider two signal-to-noise ratio (SNR) settings: i) the SNR depends on the sparsity level $K$ of input signals, and ii) the SNR is an absolute constant independent of $K$. For the first setting, we establish necessary and sufficient conditions for the exact support recovery with OMP, expressed as lower bounds on the SNR. Our results indicate that in order …
Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied in the literature. In this paper, we show that for any $K$-sparse signal $\x$, if the sensing matrix $\A$ satisfies the restricted isometry property (RIP) of order $K + 1$ with restricted isometry constant (RIC) $δ_{K+1} < 1/\sqrt {K+1}$, then under some constraint on the minimum magnitude of the nonzero elements of $\x$, the OMP algorithm exactly recovers the support of $\x$ from the measurements $\y=\A\x+\v$ in $K$ iterations, where $\v$ is the noise vector. This con…
Recovery algorithms play a key role in compressive sampling (CS). Most of current CS recovery algorithms are originally designed for one-dimensional (1D) signal, while many practical signals are two-dimensional (2D). By utilizing 2D separable sampling, 2D signal recovery problem can be converted into 1D signal recovery problem so that ordinary 1D recovery algorithms, e.g. orthogonal matching pursuit (OMP), can be applied directly. However, even with 2D separable sampling, the memory usage and complexity at the decoder is still high. This paper develops a novel recovery algorithm called 2D-OMP,…
In this paper, we consider the problem of compressed sensing where the goal is to recover almost all the sparse vectors using a small number of fixed linear measurements. For this problem, we propose a novel partial hard-thresholding operator that leads to a general family of iterative algorithms. While one extreme of the family yields well known hard thresholding algorithms like ITI (Iterative Thresholding with Inversion) and HTP (Hard Thresholding Pursuit), the other end of the spectrum leads to a novel algorithm that we call Orthogonal Matching Pursuit with Replacement (OMPR). OMPR, like th…
This paper considers the exact recovery of $k$-sparse signals in the noiseless setting and support recovery in the noisy case when some prior information on the support of the signals is available. This prior support consists of two parts. One part is a subset of the true support and another part is outside of the true support. For $k$-sparse signals $\mathbf{x}$ with the prior support which is composed of $g$ true indices and $b$ wrong indices, we show that if the restricted isometry constant (RIC) $δ_{k+b+1}$ of the sensing matrix $\mathbf{A}$ satisfies \begin{eqnarray*} δ_{k+b+1}<\frac{1}{\…
