Greedy algorithm are in widespread use for sparse recovery because of its efficiency. But some evident flaws exists in most popular greedy algorithms, such as CoSaMP, which includes unreasonable demands on prior knowledge of target signal and excessive sensitivity to random noise. A new greedy algorithm called AMOP is proposed in this paper to overcome these obstacles. Unlike CoSaMP, AMOP can extract necessary information of target signal from sample data adaptively and operate normally with little prior knowledge. The recovery error of AMOP is well controlled when random noise is presented and fades away along with increase of SNR. Moreover, AMOP has good robustness on detailed setting of target signal and less dependence on structure of measurement matrix. The validity of AMOP is verified by theoretical derivation. Extensive simulation experiment is performed to illustrate the advantages of AMOP over CoSaMP in many respects. AMOP is a good candidate of practical greedy algorithm in various applications of Compressed Sensing.
Applying the theory of compressive sensing in practice always takes different kinds of perturbations into consideration. In this paper, the recovery performance of greedy pursuits with replacement for sparse recovery is analyzed when both the measurement vector and the sensing matrix are contaminated with additive perturbations. Specifically, greedy pursuits with replacement include three algorithms, compressive sampling matching pursuit (CoSaMP), subspace pursuit (SP), and iterative hard thresholding (IHT), where the support estimation is evaluated and updated in each iteration. Based on rest…
For greedy block sparse recovery where the sparsity level is unknown, we derive a stopping condition to stop the iteration process. Focused on the block orthogonal matching pursuit (BOMP) algorithm, we model the energy of residual signals at each iteration from a probabilistic perspective. At the iteration when the last supporting block is detected, the resulting energy of residual signals is supposed to suffer an obvious decrease. Based on this, we stop the iteration process when the energy of residual signals is below a given threshold. Compared with other approaches, our derived condition w…
We describe the Simple Greedy Matrix Completion Algorithm providing an efficient method for restoration of low-rank matrices from incomplete corrupted entries. We provide numerical evidences that, even in the simplest implementation, the greedy approach may increase the recovery capability of existing algorithms significantly.
Matrix recovery is the problem of recovering a low-rank matrix from a few linear measurements. Recently, this problem has gained a lot of attention as it is employed in many applications such as Netflix prize problem, seismic data interpolation and collaborative filtering. In these applications, one might access to additional prior information about the column and row spaces of the matrix. These extra information can potentially enhance the matrix recovery performance. In this paper, we propose an efficient greedy algorithm that exploits prior information in the recovery procedure. The perform…
In this work we address the problem of recovering sparse solutions to non linear inverse problems. We look at two variants of the basic problem, the synthesis prior problem when the solution is sparse and the analysis prior problem where the solution is cosparse in some linear basis. For the first problem, we propose non linear variants of the Orthogonal Matching Pursuit (OMP) and CoSamp algorithms; for the second problem we propose a non linear variant of the Greedy Analysis Pursuit (GAP) algorithm. We empirically test the success rates of our algorithms on exponential and logarithmic functio…
Recently, greedy algorithm has received much attention as a cost-effective means to reconstruct the sparse signals from compressed measurements. Much of previous work has focused on the investigation of a single candidate to identify the support (index set of nonzero elements) of the sparse signals. Well-known drawback of the greedy approach is that the chosen candidate is often not the optimal solution due to the myopic decision in each iteration. In this paper, we propose a greedy sparse recovery algorithm investigating multiple promising candidates via the tree search. Two key ingredients o…
Compressive Sensing (CS) is a new paradigm for the efficient acquisition of signals that have sparse representation in a certain domain. Traditionally, CS has provided numerous methods for signal recovery over an orthonormal basis. However, modern applications have sparked the emergence of related methods for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary, particularly due to their potential to generate different kinds of sparse representation of signals. To this end, we apply a signal space greedy method, which relies on the ability t…
We present a novel stagewise strategy for improving greedy algorithms for sparse recovery. We demonstrate its efficiency both for synthesis and analysis sparse priors, where in both cases we demonstrate its computational efficiency and competitive reconstruction accuracy. In the synthesis case, we also provide theoretical guarantees for the signal recovery that are on par with the existing perfect reconstruction bounds for the relaxation-based solvers and other sophisticated greedy algorithms.
We study here sparse recovery problems in the presence of additive noise. We analyze a thresholding version of the CoSaMP algorithm, named Thresholding Greedy Pursuit (TGP). We demonstrate that an appropriate choice of thresholding parameter, even without the knowledge of sparsity level of the signal and strength of the noise, can result in exact recovery with no false discoveries as the dimension of the data increases to infinity.
In this article, we discuss a novel greedy algorithm for the recovery of compressive sampled signals under noisy conditions. Most of the greedy recovery algorithms proposed in the literature require sparsity of the signal to be known or they estimate sparsity, for a known representation basis, from the number of measurements. These algorithms recover signals when noise level is significantly low. We propose Entropy minimization based Matching Pursuit (EMP) which has the capability to reject noise even when noise level is comparable to that of signal level. The proposed algorithm can cater to c…
We propose a novel greedy algorithm for the support recovery of a sparse signal from a small number of noisy measurements. In the proposed method, a new support index is identified for each iteration based on bit-wise maximum a posteriori (B-MAP) detection. This is optimal in the sense of detecting one of the remaining support indices, provided that all the detected indices in the previous iterations are correct. Despite its optimality, it requires an expensive complexity for computing the maximization metric (i.e., a posteriori probability of each remaining support) due to the marginalization…
In this paper, we address the sparse multiple measurement vector (MMV) problem where the objective is to recover a set of sparse nonzero row vectors or indices of a signal matrix from incomplete measurements. Ideally, regardless of the number of columns in the signal matrix, the sparsity (k) plus one measurements is sufficient for the uniform recovery of signal vectors for almost all signals, i.e., excluding a set of Lebesgue measure zero. To approach the "k+1" lower bound with computational efficiency even when the rank of signal matrix is smaller than k, we propose a greedy algorithm called …
Intel Labs（英特尔实验室）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
A reliable support detection is essential for a greedy algorithm to reconstruct a sparse signal accurately from compressed and noisy measurements. This paper proposes a novel support detection method for greedy algorithms, which is referred to as "\textit{maximum a posteriori (MAP) support detection}". Unlike existing support detection methods that identify support indices with the largest correlation value in magnitude per iteration, the proposed method selects them with the largest likelihood ratios computed under the true and null support hypotheses by simultaneously exploiting the distribu…
Greedy Pursuits are very popular in Compressed Sensing for sparse signal recovery. Though many of the Greedy Pursuits possess elegant theoretical guarantees for performance, it is well known that their performance depends on the statistical distribution of the non-zero elements in the sparse signal. In practice, the distribution of the sparse signal may not be known a priori. It is also observed that performance of Greedy Pursuits degrades as the number of available measurements decreases from a threshold value which is method dependent. To improve the performance in these situations, we intro…
This paper examines the ability of greedy algorithms to estimate a block sparse parameter vector from noisy measurements. In particular, block sparse versions of the orthogonal matching pursuit and thresholding algorithms are analyzed under both adversarial and Gaussian noise models. In the adversarial setting, it is shown that estimation accuracy comes within a constant factor of the noise power. Under Gaussian noise, the Cramer-Rao bound is derived, and it is shown that the greedy techniques come close to this bound at high SNR. The guarantees are numerically compared with the actual perform…
In this paper, we study the problem of robust phase recovery. We investigate a novel approach based on extremely quantized (one-bit) phase-less measurements and a corresponding recovery scheme. The proposed approach has surprising robustness and stability properties and, unlike currently available methods, allows to efficiently perform phase recovery from measurements affected by severe (possibly unknown) non-linear perturbations, such as distortions (e.g. clipping). Beyond robustness, we show how our approach can be used within greedy approaches based on alternating minimization. In particula…
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 consider multichannel sparse recovery problem where the objective is to find good recovery of jointly sparse unknown signal vectors from the given multiple measurement vectors which are different linear combinations of the same known elementary vectors. Many popular greedy or convex algorithms perform poorly under non-Gaussian heavy-tailed noise conditions or in the face of outliers. In this paper, we propose the usage of mixed $\ell_{p,q}$ norms on data fidelity (residual matrix) term and the conventional $\ell_{0,2}$-norm constraint on the signal matrix to promote row-sparsity. We devise …
Sign truncated matching pursuit (STrMP) algorithm is presented in this paper. STrMP is a new greedy algorithm for the recovery of sparse signals from the sign measurement, which combines the principle of consistent reconstruction with orthogonal matching pursuit (OMP). The main part of STrMP is as concise as OMP and hence STrMP is simple to implement. In contrast to previous greedy algorithms for one-bit compressed sensing, STrMP only need to solve a convex and unconstraint subproblem at each iteration. Numerical experiments show that STrMP is fast and accurate for one-bit compressed sensing c…
Recovery of sparse signals from compressed measurements constitutes an l0 norm minimization problem, which is unpractical to solve. A number of sparse recovery approaches have appeared in the literature, including l1 minimization techniques, greedy pursuit algorithms, Bayesian methods and nonconvex optimization techniques among others. This manuscript introduces a novel two stage greedy approach, called the Forward-Backward Pursuit (FBP). FBP is an iterative approach where each iteration consists of consecutive forward and backward stages. The forward step first expands the support estimate by…
In a multiple measurement vector problem (MMV), where multiple signals share a common sparse support and are sampled by a common sensing matrix, we can expect joint sparsity to enable a further reduction in the number of required measurements. While a diversity gain from joint sparsity had been demonstrated earlier in the case of a convex relaxation method using an $l_1/l_2$ mixed norm penalty, only recently was it shown that similar diversity gain can be achieved by greedy algorithms if we combine greedy steps with a MUSIC-like subspace criterion. However, the main limitation of these hybrid …
We address the problem of sparse recovery using greedy compressed sensing recovery algorithms, without explicit knowledge of the sparsity. Estimating the sparsity order is a crucial problem in many practical scenarios, e.g., wireless communications, where exact value of the sparsity order of the unknown channel may be unavailable a priori. In this paper we have proposed a new greedy algorithm, referred to as the Multiple Choice Hard Thresholding Pursuit (MCHTP), which modifies the popular hard thresholding pursuit (HTP) suitably to iteratively recover the unknown sparse vector along with the s…
We consider the problem of sparse signal recovery from 1-bit measurements. Due to the noise present in the acquisition and transmission process, some quantized bits may be flipped to their opposite states. These sign flips may result in severe performance degradation. In this study, a novel algorithm, termed HISTORY, is proposed. It consists of Hamming support detection and coefficients recovery. The HISTORY algorithm has high recovery accuracy and is robust to strong measurement noise. Numerical results are provided to demonstrate the effectiveness and superiority of the proposed algorithm.
In this paper, we introduce a sparse approximation property of order $s$ for a measurement matrix ${\bf A}$: $$\|{\bf x}_s\|_2\le D \|{\bf A}{\bf x}\|_2+ β\frac{σ_s({\bf x})}{\sqrt{s}} \quad {\rm for\ all} \ {\bf x},$$ where ${\bf x}_s$ is the best $s$-sparse approximation of the vector ${\bf x}$ in $\ell^2$, $σ_s({\bf x})$ is the $s$-sparse approximation error of the vector ${\bf x}$ in $\ell^1$, and $D$ and $β$ are positive constants. The sparse approximation property for a measurement matrix can be thought of as a weaker version of its restricted isometry property and a stronger version of …
This paper proposes a low-computational Bayesian algorithm for noisy sparse recovery (NSR), called BHT-BP. In this framework, we consider an LDPC-like measurement matrices which has a tree-structured property, and additive white Gaussian noise. BHT-BP has a joint detection-and-estimation structure consisting of a sparse support detector and a nonzero estimator. The support detector is designed under the criterion of the minimum detection error probability using a nonparametric belief propagation (nBP) and composite binary hypothesis tests. The nonzeros are estimated in the sense of linear MMSE…
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 mea…
Orthogonal matching pursuit (OMP) is a greedy algorithm popularly being used for the recovery of sparse signals. In this paper, we study the performance of OMP for support recovery of sparse signal under noise. Our analysis shows that under mild constraint on the minimum-to-average ratio of nonzero entries in the sparse signal and the signal-to-noise ratio, the OMP algorithm can recover the support of signal with an error rate that can be arbitrarily small. Our result offers an affirmative answer to the conjecture of [Wang, TSP 2015] that the error rate of support recovery via OMP has no depen…
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…
We formulate sparse support recovery as a salient set identification problem and use information-theoretic analyses to characterize the recovery performance and sample complexity. We consider a very general model where we are not restricted to linear models or specific distributions. We state non-asymptotic bounds on recovery probability and a tight mutual information formula for sample complexity. We evaluate our bounds for applications such as sparse linear regression and explicitly characterize effects of correlation or noisy features on recovery performance. We show improvements upon previ…
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 (…
Compressive sensing (CS) technologies present many advantages over other existing approaches for implementing wideband spectrum sensing in cognitive radios (CRs), such as reduced sampling rate and computational complexity. However, there are two significant challenges: 1) choosing an appropriate number of sub-Nyquist measurements, and 2) deciding when to terminate the greedy recovery algorithm that reconstructs wideband spectrum. In this paper, an autonomous compressive spectrum sensing (ACSS) framework is presented that enables a CR to automatically choose the number of measurements while gua…
The problem of recovering the sparsity pattern of a fixed but unknown vector $β^* \in \real^p based on a set of $n$ noisy observations arises in a variety of settings, including subset selection in regression, graphical model selection, signal denoising, compressive sensing, and constructive approximation. Of interest are conditions on the model dimension $p$, the sparsity index $s$ (number of non-zero entries in $β^*$), and the number of observations $n$ that are necessary and/or sufficient to ensure asymptotically perfect recovery of the sparsity pattern. This paper focuses on the informatio…
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…
In this paper, we propose a new sparse signal recovery algorithm, referred to as sparse Kalman tree search (sKTS), that provides a robust reconstruction of the sparse vector when the sequence of correlated observation vectors are available. The proposed sKTS algorithm builds on expectation-maximization (EM) algorithm and consists of two main operations: 1) Kalman smoothing to obtain the a posteriori statistics of the source signal vectors and 2) greedy tree search to estimate the support of the signal vectors. Through numerical experiments, we demonstrate that the proposed sKTS algorithm is ef…
In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random projection; and output noise, where noise enters after the random projection. Sign pattern recovery involves the estimation of sign pattern of a sparse signal. Our idea is to pretend that no noise exists and solve the noiseless $\ell_1$ problem, namely, $\min \|β\|_1 ~ s.t. ~ y=G β$ and quantizing the resulting solution. We show that the quantized solution perfect…
We consider the problem of sparse signal recovery from noisy measurements. Many of frequently used recovery methods rely on some sort of tuning depending on either noise or signal parameters. If no estimates for either of them are available, the noisy recovery problem is significantly harder. The square root LASSO and the least absolute deviation LASSO are known to be noise-blind, in the sense that the tuning parameter can be chosen independent on the noise and the signal. We generalize those recovery methods to the \hrlone{} and give a recovery guarantee once the tuning parameter is above a t…
In this paper, we study the number of measurements required to recover a sparse signal in ${\mathbb C}^M$ with $L$ non-zero coefficients from compressed samples in the presence of noise. For a number of different recovery criteria, we prove that $O(L)$ (an asymptotically linear multiple of $L$) measurements are necessary and sufficient if $L$ grows linearly as a function of $M$. This improves on the existing literature that is mostly focused on variants of a specific recovery algorithm based on convex programming, for which $O(L\log(M-L))$ measurements are required. We also show that $O(L\log(…
A major enterprise in compressed sensing and sparse approximation is the design and analysis of computationally tractable algorithms for recovering sparse, exact or approximate, solutions of underdetermined linear systems of equations. Many such algorithms have now been proven to have optimal-order uniform recovery guarantees using the ubiquitous Restricted Isometry Property (RIP). However, it is unclear when the RIP-based sufficient conditions on the algorithm are satisfied. We present a framework in which this task can be achieved; translating these conditions for Gaussian measurement matric…
Compressive sensing (CS) is a data acquisition technique that measures sparse or compressible signals at a sampling rate lower than their Nyquist rate. Results show that sparse signals can be reconstructed using greedy algorithms, often requiring prior knowledge such as the signal sparsity or the noise level. As a substitute to prior knowledge, cross validation (CV), a statistical method that examines whether a model overfits its data, has been proposed to determine the stopping condition of greedy algorithms. This paper first analyzes cross validation in a general compressive sensing framewor…
This paper addresses recovery of a kernel $\boldsymbol{h}\in \mathbb{C}^{n}$ and a signal $\boldsymbol{x}\in \mathbb{C}^{n}$ from the low-resolution phaseless measurements of their noisy circular convolution $\boldsymbol{y} = \left \rvert \boldsymbol{F}_{lo}( \boldsymbol{x}\circledast \boldsymbol{h}) \right \rvert^{2} + \boldsymbolη$, where $\boldsymbol{F}_{lo}\in \mathbb{C}^{m\times n}$ stands for a partial discrete Fourier transform ($m<n$), $\boldsymbolη$ models the noise, and $\lvert \cdot \rvert$ is the element-wise absolute value function. This problem is severely ill-posed because both …
We develop a two-part reconstruction framework for signal recovery in compressed sensing (CS), where a fast algorithm is applied to provide partial recovery in Part 1, and a CS algorithm is applied to complete the residual problem in Part 2. Partitioning the reconstruction process into two complementary parts provides a natural trade-off between runtime and reconstruction quality. To exploit the advantages of the two-part framework, we propose a Noisy-Sudocodes algorithm that performs two-part reconstruction of sparse signals in the presence of measurement noise. Specifically, we design a fast…
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}{\…
A noisy underdetermined system of linear equations is considered in which a sparse vector (a vector with a few nonzero elements) is subject to measurement. The measurement matrix elements are drawn from a Gaussian distribution. We study the information-theoretic constraints on exact support recovery of a sparse vector from the measurement vector and matrix. We compute a tight, sufficient condition that is applied to ergodic wide-sense stationary sparse vectors. We compare our results with the existing bounds and recovery conditions. Finally, we extend our results to approximately sparse signal…
Existing approaches to compressive sensing of frequency-sparse signals focuses on signal recovery rather than spectral estimation. Furthermore, the recovery performance is limited by the coherence of the required sparsity dictionaries and by the discretization of the frequency parameter space. In this paper, we introduce a greedy recovery algorithm that leverages a band-exclusion function and a polar interpolation function to address these two issues in spectral compressive sensing. Our algorithm is geared towards line spectral estimation from compressive measurements and outperforms most exis…
We consider the Orthogonal Least-Squares (OLS) algorithm for the recovery of a $m$-dimensional $k$-sparse signal from a low number of noisy linear measurements. The Exact Recovery Condition (ERC) in bounded noisy scenario is established for OLS under certain condition on nonzero elements of the signal. The new result also improves the existing guarantees for Orthogonal Matching Pursuit (OMP) algorithm. In addition, This framework is employed to provide probabilistic guarantees for the case that the coefficient matrix is drawn at random according to Gaussian or Bernoulli distribution where we e…
Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear operator; an important special case is random design regression. We derive sharp rates of estimation for arbitrary compact elliptical parameter sets and demonstrate how they depend on the distribution of the random linear operator. Our main result is a functional that characterizes the minimax rate of estimation in terms of the noise level, the law of the r…
Secure sketch produces public information of its input $w$ without revealing it, yet, allows the exact recovery of $w$ given another value $w'$ that is close to $w$. Therefore, it can be used to reliably reproduce any error-prone secret (i.e., biometrics) stored in secret storage. However, some sources have lower entropy compared to the error itself, formally called "more error than entropy", a standard secure sketch cannot show its security promise perfectly to these kind of sources. This paper focuses on secure sketch. We propose a concrete construction for secure sketch. We show security to…
This paper addresses compressive sensing for multi-channel ECG. Compared to the traditional sparse signal recovery approach which decomposes the signal into the product of a dictionary and a sparse vector, the recently developed cosparse approach exploits sparsity of the product of an analysis matrix and the original signal. We apply the cosparse Greedy Analysis Pursuit (GAP) algorithm for compressive sensing of ECG signals. Moreover, to reduce processing time, classical signal-channel GAP is generalized to the multi-channel GAP algorithm, which simultaneously reconstructs multiple signals wit…
We consider learning Ising tree models when the observations from the nodes are corrupted by independent but non-identically distributed noise with unknown statistics. Katiyar et al. (2020) showed that although the exact tree structure cannot be recovered, one can recover a partial tree structure; that is, a structure belonging to the equivalence class containing the true tree. This paper presents a systematic improvement of Katiyar et al. (2020). First, we present a novel impossibility result by deriving a bound on the necessary number of samples for partial recovery. Second, we derive a sign…
