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146
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
 California Institute of Technology, Pasadena
, 2008
"... Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery alg ..."
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Cited by 346 (4 self)
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Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimizationbased approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(N log 2 N), where N is the length of the signal. 1.
A Simple Proof of the Restricted Isometry Property for Random Matrices
 CONSTR APPROX
, 2008
"... We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmical ..."
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Cited by 302 (57 self)
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We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finitedimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsityinducing basis.
Deterministic constructions of compressed sensing matrices
 List of References November 2001 B.1 Complete IDL Listing OMG IDL B // File: CosActivity
, 2007
"... With high esteem to Professor Henryk Wozniakowski on the occasion of his 60th birthday Compressed sensing is a new area of signal processing. Its goal is to minimize the number of samples that need to be taken from a signal for faithful reconstruction. The performance of compressed sensing on signa ..."
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Cited by 89 (2 self)
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With high esteem to Professor Henryk Wozniakowski on the occasion of his 60th birthday Compressed sensing is a new area of signal processing. Its goal is to minimize the number of samples that need to be taken from a signal for faithful reconstruction. The performance of compressed sensing on signal classes is directly related to Gelfand widths. Similar to the deeper constructions of optimal subspaces in Gelfand widths, most sampling algorithms are based on randomization. However, for possible circuit implementation, it is important to understand what can be done with purely deterministic sampling. In this note we show how to construct sampling matrices using finite fields. One such construction gives cyclic matrices which are interesting for circuit implementation. While the guaranteed performance of these deterministic constructions is not comparable to the random constructions, these matrices have the best known performance for purely deterministic constructions. 1
COMBINING GEOMETRY AND COMBINATORICS: A UNIFIED APPROACH TO SPARSE SIGNAL RECOVERY
"... Abstract. There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix Φ and then uses linear programming to decode information about x from Φx. The combinatorial approach constru ..."
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Cited by 77 (12 self)
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Abstract. There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix Φ and then uses linear programming to decode information about x from Φx. The combinatorial approach constructs Φ and a combinatorial decoding algorithm to match. We present a unified approach to these two classes of sparse signal recovery algorithms. The unifying elements are the adjacency matrices of highquality unbalanced expanders. We generalize the notion of Restricted Isometry Property (RIP), crucial to compressed sensing results for signal recovery, from the Euclidean norm to the ℓp norm for p ≈ 1, and then show that unbalanced expanders are essentially equivalent to RIPp matrices. From known deterministic constructions for such matrices, we obtain new deterministic measurement matrix constructions and algorithms for signal recovery which, compared to previous deterministic algorithms, are superior in either the number of measurements or in noise tolerance. 1.
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
, 2009
"... Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, alt ..."
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Cited by 68 (13 self)
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Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations.
Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
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Cited by 63 (13 self)
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These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
One sketch for all: Fast algorithms for compressed sensing
 In Proc. 39th ACM Symp. Theory of Computing
, 2007
"... Compressed Sensing is a new paradigm for acquiring the compressible signals that arise in many applications. These signals can be approximated using an amount of information much smaller than the nominal dimension of the signal. Traditional approaches acquire the entire signal and process it to extr ..."
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Cited by 62 (11 self)
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Compressed Sensing is a new paradigm for acquiring the compressible signals that arise in many applications. These signals can be approximated using an amount of information much smaller than the nominal dimension of the signal. Traditional approaches acquire the entire signal and process it to extract the information. The new approach acquires a small number of nonadaptive linear measurements of the signal and uses sophisticated algorithms to determine its information content. Emerging technologies can compute these general linear measurements of a signal at unit cost per measurement. This paper exhibits a randomized measurement ensemble and a signal reconstruction algorithm that satisfy four requirements: 1. The measurement ensemble succeeds for all signals, with high probability over the random choices in its construction. 2. The number of measurements of the signal is optimal, except for a factor polylogarithmic in the signal length. 3. The running time of the algorithm is polynomial in the amount of information in the signal and polylogarithmic in the signal length. 4. The recovery algorithm offers the strongest possible type of error guarantee. Moreover, it is a fully polynomial approximation scheme with respect to this type of error bound. Emerging applications demand this level of performance. Yet no other algorithm in the literature simultaneously achieves all four of these desiderata.
Computational methods for sparse solution of linear inverse problems
, 2009
"... The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, ..."
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Cited by 61 (0 self)
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The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a wealth of applications.
Sampling theorems for signals from the union of finitedimensional linear subspaces
 IEEE Trans. on Inform. Theory
, 2009
"... Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampli ..."
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Cited by 53 (8 self)
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Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the NyquistShannon sampling theorem in that they connect the number of required samples to certain model parameters. Contrary to the NyquistShannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically
Stability results for random sampling of sparse trigonometric polynomials
, 2006
"... Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that ..."
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Cited by 47 (17 self)
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Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that recovery both by a BP variant and by OMP is stable under perturbation of the samples values by noise. For BP in addition, the stability result is extended to (nonsparse) trigonometric polynomials that can be wellapproximated by sparse ones. The theoretical findings are illustrated by numerical experiments. Key Words: random sampling, trigonometric polynomials, Orthogonal Matching Pursuit, Basis Pursuit, compressed sensing, stability under noise, fast Fourier transform, nonequispaced