On Weak Tail Domination of Random Vectors

Latała, Rafał

doi:10.4064/ba57-1-8

Cited by 11 publications

(5 citation statements)

References 10 publications

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“…We will use the following comparison theorem from [20] (see also Theorem 2.5 in [10]), which is based on earlier work in [5]. It will allow us to reduce the general case of matrices with i.i.d.…”

Section: Generalization To Different Distributionsmentioning

confidence: 99%

On the Gap Between Restricted Isometry Properties and Sparse Recovery Conditions

Dirksen

Lecué

Rauhut

2018

IEEE Trans. Inform. Theory

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We consider the problem of recovering sparse vectors from underdetermined linear measurements via pconstrained basis pursuit. Previous analyses of this problem based on generalized restricted isometry properties have suggested that two phenomena occur if p = 2. First, one may need substantially more than s log(en/s) measurements (optimal for p = 2) for uniform recovery of all s-sparse vectors. Second, the matrix that achieves recovery with the optimal number of measurements may not be Gaussian (as for p = 2). We present a new, direct analysis which shows that in fact neither of these phenomena occur. Via a suitable version of the null space property we show that a standard Gaussian matrix provides q / 1-recovery guarantees for p-constrained basis pursuit in the optimal measurement regime. Our result extends to several heavier-tailed measurement matrices. As an application, we show that one can obtain a consistent reconstruction from uniform scalar quantized measurements in the optimal measurement regime.Index Terms-Restricted isometry property, compressive sensing, p-constrained basis pursuit, Gaussian random matrix, quantized compressive sensing.

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Section: Generalization To Different Distributionsmentioning

confidence: 99%

On the Gap Between Restricted Isometry Properties and Sparse Recovery Conditions

Dirksen

Lecué

Rauhut

2018

IEEE Trans. Inform. Theory

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“…The first shows that if a Bernoulli vector Y weakly dominates random vector X then Y strongly dominates X (cf. [8]).…”

Section: Selected Applicationsmentioning

confidence: 99%

On the suprema of Bernoulli processes

Latała

2013

Comptes Rendus. Mathématique

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“…By the Bobkov-Nazarov Theorem [7] they are (κ, L)-weakly dominated by the vector Y = (y 1 , ..., y n ), and κ and L are absolute constants. In [18], Lata la showed that as in (1.6), for every…”

Section: Introductionmentioning

confidence: 99%

On generic chaining and the smallest singular value of random matrices with heavy tails

Mendelson

Paouris

2012

Journal of Functional Analysis

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We present a very general chaining method which allows one to control the supremum of the empirical process sup h∈H |N −1 N i=1 h 2 (X i )− Eh 2 | in rather general situations. We use this method to establish two main results. First, a quantitative (non asymptotic) version of the classical Bai-Yin Theorem on the singular values of a random matrix with i.i.d entries that have heavy tails, and second, a sharp estimate on the quadratic empirical process when H = { t, · : t ∈ T }, T ⊂ R n and µ is an isotropic, unconditional, log-concave measure.

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On Weak Tail Domination of Random Vectors

Cited by 11 publications

References 10 publications

On the Gap Between Restricted Isometry Properties and Sparse Recovery Conditions

On the Gap Between Restricted Isometry Properties and Sparse Recovery Conditions

On the suprema of Bernoulli processes

On generic chaining and the smallest singular value of random matrices with heavy tails

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