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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 160 (19 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.
Highresolution radar via compressed sensing,” to appear
 IEEE Trans. Sign. Proc
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Sensing by Random Convolution
 IEEE Int. Work. on Comp. Adv. MultiSensor Adaptive Proc., CAMPSAP
, 2007
"... Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in a ..."
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Cited by 110 (8 self)
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Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in any fixed representation can be recovered from m � S log n measurements. We discuss two imaging scenarios — radar and Fourier optics — where convolution with a random pulse allows us to seemingly superresolve finescale features, allowing us to recover highresolution signals from lowresolution measurements. 1. Introduction. The new field of compressive sensing (CS) has given us a fresh look at data acquisition, one of the fundamental tasks in signal processing. The message of this theory can be summarized succinctly [7, 8, 10, 15, 32]: the number of measurements we need to reconstruct a signal depends on its sparsity rather than its bandwidth. These measurements, however, are different than the samples that
Toeplitz compressed sensing matrices with applications to sparse channel estimation,” submitted
 Online]. Available: http://www.ece.wisc.edu/ ∼nowak/sub08 toep.pdf
"... Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of highdimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entri ..."
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Cited by 95 (13 self)
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Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of highdimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zeromean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, timeinvariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that timedomain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional leastsquares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies. Index Terms circulant matrices, compressed sensing, Hankel matrices, restricted isometry property, sparse channel estimation, Toeplitz matrices, wireless communications. I.
Compressed channel sensing
 in Proc. of Conf. on Information Sciences and Systems (CISS
, 2008
"... Abstract—Reliable wireless communications often requires accurate knowledge of the underlying multipath channel. This typically involves probing of the channel with a known training waveform and linear processing of the input probe and channel output to estimate the impulse response. Many realworld ..."
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Cited by 52 (11 self)
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Abstract—Reliable wireless communications often requires accurate knowledge of the underlying multipath channel. This typically involves probing of the channel with a known training waveform and linear processing of the input probe and channel output to estimate the impulse response. Many realworld channels of practical interest tend to exhibit impulse responses characterized by a relatively small number of nonzero channel coefficients. Conventional linear channel estimation strategies, such as the least squares, are illsuited to fully exploiting the inherent lowdimensionality of these sparse channels. In contrast, this paper proposes sparse channel estimation methods based on convex/linear programming. Quantitative error bounds for the proposed schemes are derived by adapting recent advances from the theory of compressed sensing. The bounds come within a logarithmic factor of the performance of an ideal channel
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
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Cited by 46 (12 self)
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Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
Approximation Accuracy, Gradient Methods, and Error Bound for Structured Convex Optimization
, 2009
"... Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this ..."
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Cited by 34 (1 self)
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Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this is an error bound for the linear convergence analysis of firstorder gradient methods for solving these problems. Example applications include compressed sensing, variable selection in regression, TVregularized image denoising, and sensor network localization.
Why Gabor frames? Two fundamental measures of coherence and their role in model selection
 J. Commun. Netw
, 2010
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Suprema of chaos processes and the restricted isometry property
 Comm. Pure Appl. Math
"... We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and timefrequency structured ..."
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Cited by 32 (7 self)
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We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and timefrequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m � s log 2 s log 2 n. Key words. Compressive sensing, restricted isometry property, structured random matrices, chaos processes, γ2functionals, generic chaining, partial random circulant matrices, random Gabor synthesis matrices.
Dictionary Identification  Sparse MatrixFactorisation via ℓ1Minimisation
, 2009
"... This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via ℓ1minimisation. The problem can also be seen as factorising a d×N matrix Y = (y1... yN), yn ∈ R d of training signals into a d×K dictionary matrix Φ and a K ×N coefficient matrix ..."
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Cited by 27 (4 self)
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This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via ℓ1minimisation. The problem can also be seen as factorising a d×N matrix Y = (y1... yN), yn ∈ R d of training signals into a d×K dictionary matrix Φ and a K ×N coefficient matrix X = (x1... xN), xn ∈ R K, which is sparse. The exact question studied here is when a dictionary coefficient pair (Φ, X) can be recovered as local minimum of a (nonconvex) ℓ1criterion with input Y = ΦX. First, for general dictionaries and coefficient matrices, algebraic conditions ensuring local identifiability are derived, which are then specialised to the case when the dictionary is a basis. Finally, assuming a random BernoulliGaussian sparse model on the coefficient matrix, it is shown that sufficiently incoherent bases are locally identifiable with high probability. The perhaps surprising result is that the typically sufficient number of training samples N grows up to a logarithmic factor only linearly with the signal dimension, i.e. N ≈ CK log K, in contrast to previous approaches requiring combinatorially many samples.