Suprema of Chaos Processes and the Restricted Isometry Property

Krahmer, Felix; Mendelson, Shahar; Rauhut, Holger

doi:10.1002/cpa.21504

Cited by 221 publications

(289 citation statements)

References 49 publications

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“…Thus, the proposed scheme can be used as a hardware-friendly, memory-efficient and fast computable solution for large scale CS applications, e.g., hyperspectral imaging. During the revision of this paper, we learnt of the work [23] which presented the same bounds for the partial circulant operator; however, the bounds in [23] hold in the time domain only.…”

Section: Connections With Existing Workmentioning

confidence: 96%

Convolutional Compressed Sensing Using Deterministic Sequences

Liu

Ling

2013

IEEE Trans. Signal Process.

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Abstract-In this paper, a new class of circulant matrices built from deterministic sequences is proposed for convolutionbased compressed sensing (CS). In contrast to random convolution, the coefficients of the underlying filter are given by the discrete Fourier transform of a deterministic sequence with good autocorrelation. Both uniform recovery and non-uniform recovery of sparse signals are investigated, based on the coherence parameter of the proposed sensing matrices. Many examples of the sequences are investigated, particularly the Frank-ZadoffChu (FZC) sequence, the m-sequence and the Golay sequence. A salient feature of the proposed sensing matrices is that they can not only handle sparse signals in the time domain, but also those in the frequency and/or or discrete-cosine transform (DCT) domain.

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Section: Connections With Existing Workmentioning

confidence: 96%

Convolutional Compressed Sensing Using Deterministic Sequences

Liu

Ling

2013

IEEE Trans. Signal Process.

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“…We do not give a precise definition of this quantity (see [15,28] for more information), but recall that it can be estimated by an entropy integral (see e.g. [37, Section 1.2])…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

“…The most important ingredient for our proof is the following tail bound for suprema of second order chaos processes from [28]. We state here a sharpened version from [15,Theorem 6.5].…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

“…We establish this result by showing that a subgaussian matrix satisfies the fusion restricted isometry property with high probability. Our proof of the latter result relies heavily on a recent tail bound for suprema of second order chaos processes [28]. In the final section of the paper we give a lower bound on λ and use this to compare a necessary condition on the number of measurements needed for uniform sparse recovery via mixed ℓ 1 /ℓ 2 -minimization with the sufficient condition in Theorem 3.2.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Note that here and throughout we have used the notation u ≈ v (analogously u v) to indicate that there exists some absolute constant C > 0 such that u = Cv (u ≤ Cv). Representative families of random matrices include randomly subsampled rows from the discrete Fourier transform [46] or from a bounded orthonormal system more generally [46], [44], [42], [43], [5], and randomly-generated circulant matrices [28]. Moreover, a matrix whose entries are independent and identical (i.i.d.)…”

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confidence: 99%

Near-optimal compressed sensing guarantees for anisotropic and isotropic total variation minimization

Needell¹,

Ward²

2013

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Consider the problem of reconstructing a multidimensional signal from partial information, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. Recently, guarantees for two-dimensional images x ∈ C linear measurements y = Ax using total variation minimization to within a factor of the best s-term approximation of its gradient. The reconstruction guarantees we provide are necessarily optimal up to polynomial factors in the spatial dimension d and a logarithmic factor in the signal dimension N d . The proof relies on bounds in approximation theory concerning the compressibility of wavelet expansions of bounded-variation functions.

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Suprema of Chaos Processes and the Restricted Isometry Property

Cited by 221 publications

References 49 publications

Convolutional Compressed Sensing Using Deterministic Sequences

Convolutional Compressed Sensing Using Deterministic Sequences

Uniform recovery of fusion frame structured sparse signals

Near-optimal compressed sensing guarantees for anisotropic and isotropic total variation minimization

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