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 applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed ℓ2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

This article introduces the concept of sensing dictionaries. It presents an alteration of greedy algorithms like thresholding or (Orthogonal) Matching Pursuit which improves their performance in finding sparse signal representations in redundant dictionaries while maintaining the same complexity. These algorithms can be split into a sensing and a reconstruction step, and the former will fail to identify correct atoms if the cumulative coherence of the dictionary is too high. We thus modify the sensing step by introducing a special sensing dictionary. The correct selection of components is then determined by the cross cumulative coherence which can be considerably lower than the cumulative coherence. We characterise the optimal sensing matrix and develop a constructive method to approximate it. Finally we compare the performance of thresholding and OMP using the original and modified algorithms.

This paper presents a novel framework of fast and efficient compressive sampling based on the new concept of structurally random matrices. The proposed framework provides four important features. (i) It is universal with a variety of sparse signals. (ii) The number of measurements required for exact reconstruction is nearly optimal. (iii) It has very low complexity and fast computation based on block processing and linear filtering. (iv) It is developed on the provable mathematical model from which we are able to quantify trade-offs among streaming capability, computation/memory requirement and quality of reconstruction. All currently existing methods only have at most three out of these four highly desired features. Simulation results with several interesting structurally random matrices under various practical settings are also presented to verify the validity of the theory as well as to illustrate the promising potential of the proposed framework. Index Terms — Fast compressive sampling, random projections, nonlinear reconstruction, structurally random matrices 1.

We consider the problem of estimating a deterministic sparse vector x0 from underdetermined measurements Ax0 + w, where w represents white Gaussian noise and A is a given deterministic dictionary. We analyze the performance of three sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal matching pursuit (OMP), and thresholding. These algorithms are shown to achieve near-oracle performance with high probability, assuming that x0 is sufficiently sparse. Our results are non-asymptotic and are based only on the coherence of A, so that they are applicable to arbitrary dictionaries. Differences in the precise conditions required for the performance guarantees of each algorithm are manifested in the observed performance at high and low signal-to-noise ratios. This provides insight on the advantages and drawbacks of ℓ1 relaxation techniques such as BPDN as opposed to greedy approaches such as OMP and thresholding.

Abstract — We conducted an extensive computational experiment, lasting multiple CPU-years, to optimally select parameters for two important classes of algorithms for finding sparse solutions of underdetermined systems of linear equations. We make the optimally tuned implementations available at sparselab.stanford.edu; they run ‘out of the box ’ with no user tuning: it is not necessary to select thresholds or know the likely degree of sparsity. Our class of algorithms includes iterative hard and soft thresholding with or without relaxation, as well as CoSaMP, subspace pursuit and some natural extensions. As a result, our optimally tuned algorithms dominate such proposals. Our notion of optimality is defined in terms of phase transitions, i.e. we maximize the number of nonzeros at which the algorithm can successfully operate. We show that the phase transition is a well-defined quantity with our suite of random underdetermined linear systems. Our tuning gives the highest transition possible within each class of algorithms. We verify by extensive computation the robustness of our recommendations to the amplitude distribution of the nonzero coefficients as well as the matrix ensemble defining the underdetermined system. Our findings include: (a) For all algorithms, the worst amplitude distribution for nonzeros is generally the constantamplitude random-sign distribution, where all nonzeros are the same amplitude. (b) Various random matrix ensembles give the same phase transitions; random partial isometries may give different transitions and require different tuning; (c) Optimally tuned subspace pursuit dominates optimally tuned CoSaMP, particularly so when the system is almost square. I.

Abstract—We consider the problem of performing sparse reconstruction of large-scale data sets, such as the image sequences acquired in dynamic MRI. Here, both conventional L1 minimization through interior point methods and Orthogonal Matching Pursuit (OMP) are not practical. Instead we present an algorithm that combines fast directional updates based around conjugate gradients with an iterative thresholding step similar to that in StOMP but based upon a weak greedy selection criterion. The algorithm can achieve OMP-like performance and the rapid convergence of StOMP but with MP-like complexity per iteration. We also discuss recovery conditions applicable to this algorithm. I.

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Abstract — This paper introduces a fast and efficient framework for practical compressive sensing. Our framework is mainly based on a novel design of Structurally Random Matrix (SRM). It is highly promising for large-scale, real-time compressive sensing applications because it can be realized as a product of simple and fast operators and thus, there is no need of storing the sensing matrix explicitly. The introduced framework is flexible and provides relevant features such as universality, block-based processing and friendliness with analog and optical domain implementation. Despite all these practical advantages, the framework can be shown to approach optimal performance, i.e. the number of measurements for exact signal reconstruction is almost minimal. Simulation results with several interesting SRM under various practical settings are also presented to verify the validity of the theory as well as to illustrate the promising potentials of the proposed framework. Index Terms — compressed sensing, compressive sensing, random projection, sparse reconstruction, fast and efficient algorithm I.

Distributed compressed sensing is the extension of compressed sampling (CS) to sensor networks. The idea is to design a CS joint decoding scheme at a central decoder (base station) that exploits the inter-sensor correlations, in order to recover the whole observations from very few number of random measurements per node. In this paper, we focus on modeling the correlations and on the design and analysis of efficient joint recovery algorithms. We show, by extending earlier results of Baron et al., 1 that a simple thresholding algorithm can exploit the full diversity offered by all channels to identify a common sparse support using a near optimal number of measurements.

Abstract Over the last decade, considerable progress has been made towards developing new signal processing methods to manage the deluge of data caused by advances in sensing, imaging, storage, and computing technologies. Most of these methods are based on a simple but fundamental observation. That is, highdimensional data sets are typically highly redundant and live on low-dimensional manifolds or subspaces. This means that the collected data can often be represented in a sparse or parsimonious way in a suitably selected finite frame. This observation has also led to the development of a new sensing paradigm, called compressed sensing, which shows that high-dimensional data sets can often be reconstructed, with high fidelity, from only a small number of measurements. Finite frames play a central role in the design and analysis of both sparse representations and compressed sensing methods. In this chapter, we highlight this role primarily in the context of compressed sensing for estimation, recovery, support detection, regression, and detection of sparse signals. The recurring theme is that frames with small spectral norm and/or small worst-case coherence, average coherence, or sum coherence are well-suited for making measurements of sparse signals. Key words: approximation theory, coherence property, compressed sensing, detection, estimation, Grassmannian frames, model selection, regression, restricted isometry