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506
Greed is Good: Algorithmic Results for Sparse Approximation
, 2004
"... This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho’s basis pursuit (BP) paradigm can recover the optimal representa ..."
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Cited by 916 (8 self)
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This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho’s basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasiincoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasiincoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.
Grassmannian frames with applications to coding and communication
 Appl. Comp. Harmonic Anal
, 2003
"... For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation 〈fk, fl〉  among all frames {fk}k∈I ∈ F. We first analyze finitedimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal sph ..."
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Cited by 229 (14 self)
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For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation 〈fk, fl〉  among all frames {fk}k∈I ∈ F. We first analyze finitedimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with unit norm tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on unit norm tight frames. We then introduce infinitedimensional Grassmannian frames and analyze their connection to unit norm tight frames for frames which are generated by grouplike unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.
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 162 (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
, 2008
"... A stylized compressed sensing radar is proposed in which the timefrequency plane is discretized into an N ×N grid. Assuming the number of targets K is small (i.e., K ≪ N 2), then we can transmit a sufficiently “incoherent ” pulse and employ the techniques of compressed sensing to reconstruct the ta ..."
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Cited by 156 (9 self)
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A stylized compressed sensing radar is proposed in which the timefrequency plane is discretized into an N ×N grid. Assuming the number of targets K is small (i.e., K ≪ N 2), then we can transmit a sufficiently “incoherent ” pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity K is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel compressed sensing approach offers great potential for better resolution over classical radar.
Shearlet Coorbit Spaces: Compactly Supported Analyzing Shearlets
 Traces and Embeddings, J. Fourier Anal. Appl
"... Abstract. We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Bes ..."
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Cited by 91 (14 self)
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Abstract. We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. Furthermore, we show embedding relations of traces of these subspaces with respect to the real axes. 1.
Wiener's Lemma For Twisted Convolution And Gabor Frames
, 2001
"... We prove noncommutative versions of Wiener's Lemma on absolutely convergent Fourier series (a) for the case of twisted convolution and (b) for rotation algebras. As an application we solve some open problems about Gabor frames, among them the problem of Feichtinger and Janssen that is known in ..."
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Cited by 85 (21 self)
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We prove noncommutative versions of Wiener's Lemma on absolutely convergent Fourier series (a) for the case of twisted convolution and (b) for rotation algebras. As an application we solve some open problems about Gabor frames, among them the problem of Feichtinger and Janssen that is known in the literature as the "irrational case".
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator
"... We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically t ..."
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Cited by 79 (9 self)
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We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about nonuniform sampling in shiftinvariant spaces. 1.
The Kadison–Singer problem in mathematics and engineering
 Proc. Natl. Acad. Sci. USA 103 (2006
, 2006
"... Abstract. We will show that the famous, intractible 1959 KadisonSinger problem in C ∗algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well ..."
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Cited by 68 (19 self)
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Abstract. We will show that the famous, intractible 1959 KadisonSinger problem in C ∗algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of KadisonSinger. In some areas we will prove what we believe will be the strongest results ever available in the case that KadisonSinger fails. Finally, we will give some directions for constructing a counterexample to KadisonSinger. 1.
TimeFrequency Analysis of Localization Operators
 J. FUNCT. ANAL
, 2002
"... We study a class of pseudodifferential operators known as timefrequency localization operators, AntiWick operators, GaborToeplitz operators or wave packets. Given a symbol a and two windows ' 1 ; ' 2 , we investigate the multilinear mapping from (a; ' 1 ; ' 2 ) 2 S ) to ..."
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Cited by 67 (29 self)
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We study a class of pseudodifferential operators known as timefrequency localization operators, AntiWick operators, GaborToeplitz operators or wave packets. Given a symbol a and two windows ' 1 ; ' 2 , we investigate the multilinear mapping from (a; ' 1 ; ' 2 ) 2 S ) to the localization operator A a and we give sufficient and necessary conditions for a to be bounded or to belong to a Schatten class. Our results are formulated in terms of timefrequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.
Density, overcompleteness, and localization of frames
 I. THEORY, J. FOURIER ANAL. APPL
, 2005
"... This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in ..."
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Cited by 63 (20 self)
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This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E viaamapa: I → G. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of F, the relative measure of E—both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements—and the density of the set a(I) inG. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. These abstract results yield an array of new implications for irregular Gabor frames. Various Nyquist density results for Gabor frames are recovered as special cases, but in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the timefrequency plane.