Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discrete-to-discrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuous-time signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
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2009-09-22

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2008-03-17
Compressive sampling offers a new paradigm for acquiring signals that arecompressible with respect to an orthonormal basis. The major algorithmicchallenge in compressive sampling is to approximate a compressible signal fromnoisy samples. This paper describes a new iterative recovery algorithm calledCoSaMP that delivers the same guarantees as the best optimization-basedapproaches. Moreover, this algorithm offers rigorous bounds on computationalcost and storage. It is likely to be extremely efficient for practical problemsbecause it requires only matrix-vector multiplies with the sampling matrix. Formany cases of interest, the running time is just O(N*log^2(N)), where N is thelength of the signal.
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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results , the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis .
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An important problem in the theory of sparse approximation is to identify well-conditioned subsets of vectors from a general dictionary. In most cases, current results do not apply unless the number of vectors is smaller than the square root of the ambient dimension, so these bounds are too weak for many applications. This paper shatters the square-root bottleneck by focusing on random subdictionaries instead of arbitrary subdictionaries. It provides explicit bounds on the extreme singular values of random subdictionaries that hold with overwhelming probability. The results are phrased in terms of the coherence and spectral norm of the dictionary, which capture information about its global geometry. The proofs rely on standard tools from the area of Banach space probability. As an application, the paper shows that the conditioning of a subdictionary is the major obstacle to the uniqueness of sparse representations and the success of 1 minimization techniques for signal recovery. Indeed, if a fixed subdictionary is well conditioned and its cardinality is slightly smaller than the ambient dimension, then a random signal formed from this subdictionary almost surely has no other representation that is equally sparse. Moreover, with overwhelming probability, the maximally sparse representation can be identified via 1 minimization. Note that the results in this paper are not directly comparable with recent work on subdictionaries of random dictionaries.
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Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then lowpass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, realtime performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.
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This paper demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m 2) measurements. The new results for OMP are comparable with recent results for another approach called Basis Pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems.
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This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F ; it includes all models-e.g. Gaussian, frequency measurements-discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP)-they make use of a much weaker notion-or a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.
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Suppose we are given a vector f in a class F ⊂ R N , e.g. a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision ǫ in the Euclidean (ℓ 2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or com-pressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector |f | (or of its coefficients in a fixed basis) obeys |f | (n) ≤ R · n −1/p , where R > 0 and p > 0. Suppose that we take measurements y k = f, X k , k = 1,. .. , K, where the X k are N-dimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0 < p < 1 and with overwhelming probability, our reconstruction f ♯ , defined as the solution to the constraints y k = f ♯ , X k with minimal ℓ 1 norm, obeys f − f ♯ ℓ2 ≤ C p · R · (K/ log N) −r , r = 1/p − 1/2. There is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of K measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes few randomly sampled Fourier coefficients of f. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed.
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2006-11-30
We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from alimited number of linear measurements. Given $m$ randomly selected samples of$U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimizationrecovers $x^0$ exactly when the number of measurements exceeds $m\geq\mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n,$ where $S$ is the number ofnonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properlynormalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$,the fewer samples needed. The result holds for most'' sparse signals $x^0$ supported on a fixed (butarbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on$T$ and the observed values of $Ux^0$ are drawn at random, the signal isrecovered with overwhelming probability. Moreover, there is a sense in whichthis is nearly optimal since any method succeeding with the same probabilitywould require just about this many samples.
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We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two. Analog compression that narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm then detects the input subspace prior to conventional signal processing. A representative union model of spectrally-sparse signals serves as a test-case to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy and software complexities. We conduct a comprehensive comparison between two sub-Nyquist acquisition strategies for spectrally-sparse signals, the random demodulator and the modulated wideband converter (MWC), in terms of these metrics and draw operative conclusions regarding the choice of analog compression. We then address lowrate signal processing and develop an algorithm for that purpose that enables convenient signal processing at sub-Nyquist rates from samples obtained by the MWC. We conclude by showing that a variety of other sampling approaches for different union classes fit nicely into our framework.
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We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finite-dimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finite-dimensional techniques are ill-suited for solving a number of important problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinite-dimensional reconstruction problem.
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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon's R(D) theory in the case of Gaussian stationary processes, which says that transforming into a Fourier basis followed by block coding gives an optimal lossy compression technique; practical developments like transform-based image compression have been inspired by this result. In this paper we also discuss connections perhaps less familiar to the Information Theory community, growing out of the field of harmonic analysis. Recent harmonic analysis constructions, such as wavelet transforms and Gabor transforms, are essentially optimal transforms for transform coding in certain settings. Some of these transforms are under consideration for future compression standards. We discuss some of the lessons of harmonic analysis in this century. Typically, the problems and achievements of this field have involved goals that were not obviously related to practical data compression, and have used a language not immediately accessible to outsiders. Nevertheless, through an extensive generalization of what Shannon called the "sampling theorem," harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations. We explain why harmonic analysis has interacted with data compression, and we describe some interesting recent ideas in the field that may affect data compression in the future.
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This work investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. Even with this continuous dictionary, it is shown that O(s log s log n) random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method.
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This article demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results for OMP, which require O(m 2) measurements. The new results for OMP are comparable with recent results for another algorithm called Basis Pursuit (BP). The OMP algorithm is much faster and much easier to implement, which makes it an attractive alternative to BP for signal recovery problems.
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We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank-1 matrix formed by their outer product wx *. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for "generic" signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N , and w is a vector in a subspace of dimension K whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers wx * without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length N which we code using a random L × N coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length K, then the receiver can recover both the channel response and the message when L N + K, to within constant and log factors.
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Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e. the number of pixels in the image. This paper surveys an emerging theory which goes by the name of "compressive sampling" or "compressed sensing," and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image. It is believed that compressive sampling has far reaching implications. For example, it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary. This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously. In this short survey, we provide some of the key mathematical insights underlying this new theory, and explain some of the interactions between compressive sampling and other fields such as statistics, information theory, coding theory, and theoretical computer science. Mathematics Subject Classification (2000). Primary 00A69, 41-02, 68P30; Secondary 62C65.
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This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffd-ing, and McDiarmid. The matrix inequalities promise the same diversity of application , ease of use, and strength of conclusion that have made the scalar inequalities so valuable.
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This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object-the high end of its spectrum-from coarse scale information only-from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0, 1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cutoff f c. We show that one can super-resolve these point sources with infinite precision-i.e. recover the exact locations and amplitudes-by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/f c. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary.
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