This work considers recovery of signals that are sparse over two bases. For instance, a signal might be sparse in both time and frequency, or a matrix can be low rank and sparse simultaneously. To facilitate recovery, we consider minimizing the sum of the $\ell_1$-norms that correspond to each basis, which is a tractable convex approach. We find novel optimality conditions which indicates a gain over traditional approaches where $\ell_1$ minimization is done over only one basis. Next, we analyze these optimality conditions for the particular case of time-frequency bases. Denoting sparsity in the first and second bases by $k_1,k_2$ respectively, we show that, for a general class of signals, using this approach, one requires as small as $O(\max\{k_1,k_2\}\log\log n)$ measurements for successful recovery hence overcoming the classical requirement of $Θ(\min\{k_1,k_2\}\log(\frac{n}{\min\{k_1,k_2\}}))$ for $\ell_1$ minimization when $k_1\approx k_2$. Extensive simulations show that, our analysis is approximately tight.
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…
A host of problems involve the recovery of structured signals from a dimensionality reduced representation such as a random projection; examples include sparse signals (compressive sensing) and low-rank matrices (matrix completion). Given the wide range of different recovery algorithms developed to date, it is natural to ask whether there exist "universal" algorithms for recovering "structured" signals from their linear projections. We recently answered this question in the affirmative in the noise-free setting. In this paper, we extend our results to the case of noisy measurements.
In many data acquisition systems it is common to observe signals whose amplitudes have been clipped. We present two new algorithms for recovering a clipped signal by leveraging the model assumption that the underlying signal is sparse in the frequency domain. Both algorithms employ ideas commonly used in the field of Compressive Sensing; the first is a modified version of Reweighted $\ell_1$ minimization, and the second is a modification of a simple greedy algorithm known as Trivial Pursuit. An empirical investigation shows that both approaches can recover signals with significant levels of cl…
The reconstruction of sparse signals requires the solution of an $\ell_0$-norm minimization problem in Compressed Sensing. Previous research has focused on the investigation of a single candidate to identify the support (index of nonzero elements) of a sparse signal. To ensure that the optimal candidate can be obtained in each iteration, we propose here an iterative greedy reconstruction algorithm (GSRA). First, the intersection of the support sets estimated by the Orthogonal Matching Pursuit (OMP) and Subspace Pursuit (SP) is set as the initial support set. Then, a hope-tree is built to expan…
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…
We consider the problem of recovering a target matrix that is a superposition of low-rank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured high-dimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant low-rank recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers low-rank and sparse terms, provided the number o…
The nascent field of compressed sensing is founded on the fact that high-dimensional signals with "simple structure" can be recovered accurately from just a small number of randomized samples. Several specific kinds of structures have been explored in the literature, from sparsity and group sparsity to low-rankness. However, two fundamental questions have been left unanswered, namely: What are the general abstract meanings of "structure" and "simplicity"? And do there exist universal algorithms for recovering such simple structured objects from fewer samples than their ambient dimension? In th…
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 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…
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…
The fast growing field of compressed sensing is founded on the fact that if a signal is 'simple' and has some 'structure', then it can be reconstructed accurately with far fewer samples than its ambient dimension. Many different plausible structures have been explored in this field, ranging from sparsity to low-rankness and to finite rate of innovation. However, there are important abstract questions that are yet to be answered. For instance, what are the general abstract meanings of 'structure' and 'simplicity'? Do there exist universal algorithms for recovering such simple structured objects…
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…
Orthogonal matching pursuit (OMP) is a widely used compressive sensing (CS) algorithm for recovering sparse signals in noisy linear regression models. The performance of OMP depends on its stopping criteria (SC). SC for OMP discussed in literature typically assumes knowledge of either the sparsity of the signal to be estimated $k_0$ or noise variance $σ^2$, both of which are unavailable in many practical applications. In this article we develop a modified version of OMP called tuning free OMP or TF-OMP which does not require a SC. TF-OMP is proved to accomplish successful sparse recovery under…
In the recent work of Candes et al, the problem of recovering low rank matrix corrupted by i.i.d. sparse outliers is studied and a very elegant solution, principal component pursuit, is proposed. It is motivated as a tool for video surveillance applications with the background image sequence forming the low rank part and the moving objects/persons/abnormalities forming the sparse part. Each image frame is treated as a column vector of the data matrix made up of a low rank matrix and a sparse corruption matrix. Principal component pursuit solves the problem under the assumptions that the singul…
This paper studies the joint user association and resource allocation in heterogeneous networks (HetNets) from a novel perspective, motivated by and generalizing the idea of fractional frequency reuse. By treating the multi-cell multi-user resource allocation as resource partitioning among multiple reuse patterns, we propose a unified framework to analyze and compare a wide range of user association and resource allocation strategies for HetNets, and provide an optimal benchmark for network performance. The enabling mechanisms are a novel formulation to consider all possible interference patte…
Exact recovery of $K$-sparse signals $x \in \mathbb{R}^{n}$ from linear measurements $y=Ax$, where $A\in \mathbb{R}^{m\times n}$ is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is widely used for reconstructing $x$. A fundamental question in the performance analysis of OMP is the characterizations of the probability of exact recovery of $x$ for random matrix $A$ and the minimal $m$ to guarantee a target recovery performance. In many practical applications, in addition to sparsity, $x$ also has some additional properties. This paper shows that…
We introduce a two step algorithm with theoretical guarantees to recover a jointly sparse and low-rank matrix from undersampled measurements of its columns. The algorithm first estimates the row subspace of the matrix using a set of common measurements of the columns. In the second step, the subspace aware recovery of the matrix is solved using a simple least square algorithm. The results are verified in the context of recovering CINE data from undersampled measurements; we obtain good recovery when the sampling conditions are satisfied.
In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block sparse signals $\x$ from measurements $\y=\A\x+\v$, where $\v$ is an $\ell_2$-bounded noise vector (i.e., $\|\v\|_2\leq ε$ for some constant $ε$). We investigate some sufficient conditions based on the block restricted isometry property (block-RIP) for exact (when $\v=\0$) and stable (when $\v\neq\0$) recovery of block sparse signals $\x$. First, on the one hand, we show that if $\A$ satisfies the block-RIP with $δ_{K+1}<1/\sqrt{K+1}$, then every block $K$-sparse signal $\x$ can be exactly or stably r…
In this paper, we put forth a new joint sparse recovery algorithm called signal space matching pursuit (SSMP). The key idea of the proposed SSMP algorithm is to sequentially investigate the support of jointly sparse vectors to minimize the subspace distance to the residual space. Our performance guarantee analysis indicates that SSMP accurately reconstructs any row $K$-sparse matrix of rank $r$ in the full row rank scenario if the sampling matrix $\mathbf{A}$ satisfies $\text{krank}(\mathbf{A}) \ge K+1$, which meets the fundamental minimum requirement on $\mathbf{A}$ to ensure exact recovery. …
We consider the problem of recovering sparse vectors from underdetermined linear measurements via $\ell_p$-constrained basis pursuit. Previous analyses of this problem based on generalized restricted isometry properties have suggested that two phenomena occur if $p\neq 2$. First, one may need substantially more than $s \log(en/s)$ measurements (optimal for $p=2$) for uniform recovery of all $s$-sparse vectors. Second, the matrix that achieves recovery with the optimal number of measurements may not be Gaussian (as for $p=2$). We present a new, direct analysis which shows that in fact neither o…
We consider high dimensional sparse regression, and develop strategies able to deal with arbitrary -- possibly, severe or coordinated -- errors in the covariance matrix $X$. These may come from corrupted data, persistent experimental errors, or malicious respondents in surveys/recommender systems, etc. Such non-stochastic error-in-variables problems are notoriously difficult to treat, and as we demonstrate, the problem is particularly pronounced in high-dimensional settings where the primary goal is {\em support recovery} of the sparse regressor. We develop algorithms for support recovery in s…
Given an overcomplete dictionary $A$ and a signal $b$ that is a linear combination of a few linearly independent columns of $A$, classical sparse recovery theory deals with the problem of recovering the unique sparse representation $x$ such that $b = A x$. It is known that under certain conditions on $A$, $x$ can be recovered by the Basis Pursuit (BP) and the Orthogonal Matching Pursuit (OMP) algorithms. In this work, we consider the more general case where $b$ lies in a low-dimensional subspace spanned by some columns of $A$, which are possibly linearly dependent. In this case, the sparsest s…
Laboratory for Information Inference Systems \' E cole Polytechnique F\' e d\' e rale de Lausanne
The standard approach to compressive sampling considers recovering an unknown deterministic signal with certain known structure, and designing the sub-sampling pattern and recovery algorithm based on the known structure. This approach requires looking for a good representation that reveals the signal structure, and solving a non-smooth convex minimization problem (e.g., basis pursuit). In this paper, another approach is considered: We learn a good sub-sampling pattern based on available training signals, without knowing the signal structure in advance, and reconstruct an accordingly sub-sample…
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…
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 …
In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to a…
In this paper we revisit the sparse multiple measurement vector (MMV) problem where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem has received increasing interest as an extension of the single channel sparse recovery problem which lies at the heart of the emerging field of compressed sensing. However the sparse approximation problem has origins which include links to the field of array signal processing where we find the inspiration for a new family of MMV algorithms based on the MUSIC algorithm. We highlight the role of the rank …
In compressed sensing problems, $\ell_1$ minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical interest to find alternative polynomial-time algorithms which perform better than $\ell_1$ minimization. \cite{Icassp reweighted l_1}, \cite{Isit reweighted l_1}, \cite{XuScaingLaw} and \cite{iterativereweightedjournal} have shown that a two-stage re-weighted $\ell_1$ minimization algorithm can boost the phase transition performance for signals whose nonzero elem…
We propose and analyze a solution to the problem of recovering a block sparse signal with sparse blocks from linear measurements. Such problems naturally emerge inter alia in the context of mobile communication, in order to meet the scalability and low complexity requirements of massive antenna systems and massive machine-type communication. We introduce a new variant of the Hard Thresholding Pursuit (HTP) algorithm referred to as HiHTP. We provide both a proof of convergence and a recovery guarantee for noisy Gaussian measurements that exhibit an improved asymptotic scaling in terms of the sa…
Many applications concern sparse signals, for example, detecting anomalies from the differences between consecutive images taken by surveillance cameras. This paper focuses on the problem of recovering a K-sparse signal x in N dimensions. In the mainstream framework of compressed sensing (CS), the vector x is recovered from M non-adaptive linear measurements y = xS, where S (of size N x M) is typically a Gaussian (or Gaussian-like) design matrix, through some optimization procedure such as linear programming (LP). In our proposed method, the design matrix S is generated from an $α$-stable dist…
