Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles

Mendelson, Shahar; Pajor, Alain; Tomczak-Jaegermann, Nicole

doi:10.1007/s00365-007-9005-8

Cited by 231 publications

(204 citation statements)

References 9 publications

(32 reference statements)

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“…where C is a given constant, then δ k ≤ ǫ k with high probability [7,9,18]. A similar result for the same family of random matrices holds for the D-RIP [14].…”

Section: Algorithm Guaranteessupporting

confidence: 55%

Greedy signal space methods for incoherence and beyond

Giryes

Needell

2015

Applied and Computational Harmonic Analysis

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ABSTRACT. Compressive sampling (CoSa) has provided many methods for signal recovery of signals compressible with respect to an orthonormal basis. However, modern applications have sparked the emergence of approaches for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary. Recently, several "signal-space" greedy methods have been proposed to address signal recovery in this setting. However, such methods inherently rely on the existence of fast and accurate projections which allow one to identify the most relevant atoms in a dictionary for any given signal, up to a very strict accuracy. When the dictionary is highly overcomplete, no such projections are currently known; the requirements on such projections do not even hold for incoherent or well-behaved dictionaries. In this work, we provide an alternate analysis for signal space greedy methods which enforce assumptions on these projections which hold in several settings including those when the dictionary is incoherent or structurally coherent. These results align more closely with traditional results in the standard CoSa literature and improve upon previous work in the signal space setting.

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“…where C is a given constant, then δ k ≤ ǫ k with high probability [7,9,18]. A similar result for the same family of random matrices holds for the D-RIP [14].…”

Section: Algorithm Guaranteessupporting

confidence: 55%

Greedy signal space methods for incoherence and beyond

Giryes

Needell

2015

Applied and Computational Harmonic Analysis

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“…For example, Gaussian random matrices, that is, matrices that have independent, normally distributed entries with mean zero and variance one, have been shown [3,13,34] to have restricted isometry constants of 1 √ n A satisfy δ s ≤ δ with high probability provided that n ≥ Cδ −2 s log(N/s).…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…See [3,13,34,21,44] for precise statements and extensions to Bernoulli and subgaussian matrices. It follows from lower estimates of Gelfand widths that this bound on the required samples is optimal [17,25,26], that is, the log-factor must be present.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

The restricted isometry property for time–frequency structured random matrices

Pfander

Rauhut

Tropp

2012

Probab. Theory Relat. Fields

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We establish the restricted isometry property for finite dimensional Gabor systems, that is, for families of time-frequency shifts of a randomly chosen window function. We show that the s-th order restricted isometry constant of the associated n × n 2 Gabor synthesis matrix is small provided s ≤ c n 2/3 / log 2 n. This improves on previous estimates that exhibit quadratic scaling of n in s. Our proof develops bounds for a corresponding chaos process.

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“…The linear measurement process is typically described via random matrices. For instance, Gaussian random matrices provide optimal recovery guarantees in the sense that m ≥ Cs log(N/s) measurements are necessary and sufficient to recover any s-sparse vector in dimension N via ℓ 1 -minimization and other recovery algorithms [8,31].…”

Section: Introductionmentioning

confidence: 99%

Uniform recovery of fusion frame structured sparse signals

Ayaz

Dirksen

Rauhut

2016

Applied and Computational Harmonic Analysis

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We consider the problem of recovering fusion frame sparse signals from incomplete measurements. These signals are composed of a small number of nonzero blocks taken from a family of subspaces. First, we show that, by using a-priori knowledge of a coherence parameter associated with the angles between the subspaces, one can uniformly recover fusion frame sparse signals with a significantly reduced number of vector-valued (sub-)Gaussian measurements via mixed ℓ 1 /ℓ 2 -minimization. We prove this by establishing an appropriate version of the restricted isometry property. Our result complements previous nonuniform recovery results in this context, and provides stronger stability guarantees for noisy measurements and approximately sparse signals. Second, we determine the minimal number of scalar-valued measurements needed to uniformly recover all fusion frame sparse signals via mixed ℓ 1 /ℓ 2 -minimization. This bound is achieved by scalar-valued subgaussian measurements. In particular, our result shows that the number of scalar-valued subgaussian measurements cannot be further reduced using knowledge of the coherence parameter. As a special case it implies that the best known uniform recovery result for block sparse signals using subgaussian measurements is optimal.

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Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles

Cited by 231 publications

References 9 publications

Greedy signal space methods for incoherence and beyond

Greedy signal space methods for incoherence and beyond

The restricted isometry property for time–frequency structured random matrices

Uniform recovery of fusion frame structured sparse signals

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